Class 12th Mathematics Part Ii CBSE Solution
Exercise 9.1- d^4y/dx^4 + sin (y^there there eξ sts there there eξ sts there there eξ sts) = 0…
- y′ + 5y = 0 Determine order and degree (if defined) of differential equations…
- (ds/dt)^4 + 3s d^2s/dt^2 = 0 Determine order and degree (if defined) of…
- (d^2y/dx^2)^2 + cos (dy/dx) = 0 Determine order and degree (if defined) of…
- d^2y/dx^2 = cos3x+sin3x Determine order and degree (if defined) of differential…
- (y″′)^2 + (y″)^3 + (y′)^4 + y^5 = 0 Determine order and degree (if defined) of…
- y″′ +2y” + y’ = 0 Determine order and degree (if defined) of differential…
- y′ + y = ex Determine order and degree (if defined) of differential equations…
- y″ + (y′) + 2y = 0 Determine order and degree (if defined) of differential…
- y″ + 2y′ + sin y = 0 Determine order and degree (if defined) of differential…
- The degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 + sin (dy/dx)…
- The order of the differential equation 2x^2 d^2y/dx^2 - 3 dy/dx + y = 0 isA. 2…
Exercise 9.2- y = ex + 1 : y′′ - y′ = 0 In each of the question verify that the given…
- y = x^2 + 2x + C : y′ - 2x - 2 = 0 In each of the question verify that the given…
- y = cos x + C : y′ + sin x = 0 In each of the question verify that the given…
- y = root 1+x^2 : y^there there eξ sts = xy/1+x^2 In each of the question verify…
- y = Ax : xy′ = y (x ≠ 0) In each of the question verify that the given functions…
- y = x sin x : xy′ = y + x root x^2 - y^2 (x ≠ 0 and x y or x - y) In each of the…
- xy = log y + C : y^there there eξ sts = y^2/1-xy (xy not equal 1) In each of the…
- y - cos y = x : (y sin y + cos y + x)y′ = y In each of the question verify that…
- x + y = tan-1y : y^2 y′ + y^2 + 1 = 0 In each of the question verify that the…
- y = root a^2 - x^2 x in (- a , a) :x+y dy/dx = 0 (y not equal 0) In each of the…
- The number of arbitrary constants in the general solution of a differential…
- The number of arbitrary constants in the particular solution of a differential…
Exercise 9.3- x/a + y/b = 1 In each of the question, form a differential equation representing…
- y^2 = a(b^2 - x^2) In each of the question, form a differential equation…
- y = a e3x + b e-2x In each of the question, form a differential equation…
- y = e2x (a + bx) In each of the question, form a differential equation…
- y = ex (a cos x + b sin x) In each of the question, form a differential equation…
- Form the differential equation of the family of circles touching the y-axis at…
- Form the differential equation of the family of parabolas having vertex at…
- Form the differential equation of the family of ellipses having foci on y-axis…
- Form the differential equation of the family of hyperbolas having foci on x-axis…
- Form the differential equation of the family of circles having centre on y-axis…
- Which of the following differential equations has y = c1ex + c2e-x as the…
- Which of the following differential equations has y = x as one of its…
Exercise 9.4- dy/dx = 1-cosx/1+cosx For each of the differential equations in question, find…
- dy/dx = root 4-y^2 (-2y2) For each of the differential equations in question,…
- dy/dx + y = 1 (y not equal 1) For each of the differential equations in…
- sec^2 x tan y dx + sec^2 y tan x dy For each of the differential equations in…
- (ex + e-x)dy - (ex - e-x)dx = 0 For each of the differential equations in…
- dy/dx = (1+x^2) (1+y^2) For each of the differential equations in question, find…
- y log y dx - x dy = 0 For each of the differential equations in question, find…
- x^5 dy/dx = - y^5 For each of the differential equations in question, find the…
- dy/dx = sin^-1x For each of the differential equations in question, find the…
- extan y dx + 1(1 - ex)sec^2 y dy = 0 For each of the differential equations in…
- (x^3 + x^2 + x + 1) dy/dx = 2x^2 + x, y = 1 when x = 0 For each of the…
- x(x^2 - 1) dy/dx = 1; y = 0 when x = 0 For each of the differential equations…
- cos dy/dx = a; y = 2 when x = 0 For each of the differential equations in…
- dy/dx = y tan x; y = 1 when x = 0 For each of the differential equations in…
- Find the equation of a curve passing through the point (0, 0) and whose…
- For the differential equation xy dy/dx = (x+2) (y+2) find the solution curve…
- Find the equation of a curve passing through the point (0, -2) given that at…
- At any point (x, y) of a curve, the slope of the tangent is twice the slope of…
- The volume of spherical balloon being inflated changes at a constant rate. If…
- In a bank, principal increases continuously at the rate of r% per year. Find…
- In a bank, principal increases continuously at the rate of 5% per year. An…
- In a culture, the bacteria count is 1,00,000. The number is increased by 10% in…
- The general solution of the differential equation dy/dx = e^x+y isA. ex + e-y =…
Exercise 9.5- (x^2 + xy)dy = (x^2 + y^2)dx In each of the question, show that the given…
- y^there there eξ sts = x+y/x In each of the question, show that the given…
- (x - y)dy - (x + y)dx = 0 In each of the question, show that the given…
- (x^2 - y^2)dx + 2xy dy = 0 In each of the question, show that the given…
- x^2 dy/dx = x^2 - 2y^2 + xy In each of the question, show that the given…
- xdy-ydx = root x^2 + y^2 dx In each of the question, show that the given…
- xcos (y/x) + ysin (y/x) ydx = ysin (y/x) - xcos (y/x) xdy In each of the…
- x dy/dx - y+xsin (y/x) = 0 In each of the question, show that the given…
- ydx+xlog (y/x) dy-2xdy = 0 In each of the question, show that the given…
- (1+e^x/y) dx+e^x/y (1 - x/y) dy = 0 In each of the question, show that the…
- (x + y)dy + (x - y) dx = 0; y = 1 when x = 1 For each of the differential…
- x^2 dy + (xy + y^2)dx = 0; y = 1 when x = 1 For each of the differential…
- [xsin^2 (y/x) - y]dx+xdy = 0 y = pi /4 whenx = 1 For each of the differential…
- dy/dx - y/x + cosec (y/x) = 0 y = 0 when x = 1 For each of the differential…
- 2xy+y^2 - 2x^2 dy/dx = 0 y = 2whenx = 1 For each of the differential equations…
- A homogeneous differential equation of the from dx/dy = h (x/y) can be solved…
- A homogeneous differential equation of the from dx/dy = h (x/y) can be solved…
Exercise 9.6- dy/dx + 2y = sinx For each of the differential equations given in question, find…
- dy/dx + 3y = e^-2x For each of the differential equations given in question,…
- dy/dx + y/x = x^2 For each of the differential equations given in question, find…
- dy/dx + (secx) y = tanx (0 less than equal to x pi /2) For each of the…
- cos^2x dy/dx + y = tanx (0 less than equal to x pi /2) For each of the…
- x dy/dx + 2y = x^2logx For each of the differential equations given in question,…
- xlogx dy/dx + y = 2/x logx For each of the differential equations given in…
- (1 + x^2)dy + 2xy dx = cot x dx (x ≠ 0) For each of the differential equations…
- x dy/dx + y-x+xycotx = 0 (x not equal 0) For each of the differential equations…
- (x+y) dy/dx = 1 For each of the differential equations given in question, find…
- y dx + (x - y^2)dy = 0 For each of the differential equations given in…
- (x+3y^2) dy/dx = y (y0) For each of the differential equations given in…
- dy/dx + 2ytanx = sinxy = 0 x = pi /3 For each of the differential equations…
- (1+x^2) dy/dx + 2xy = 1/1+x^2 y = 0 x = 1 For each of the differential…
- dy/dx - 3ycotx = sin2xy = 2 x = pi /2 For each of the differential equations…
- Find the equation of a curve passing through the origin given that the slope of…
- Find the equation of a curve passing through the point (0, 2) given that the…
- The Integrating Factor of the differential equation x dy/dx - y = 2x^2 isA. e-x…
- The Integrating Factor of the differential equation (1-y^2) dx/dy + yx = ay…
Miscellaneous Exercise- d^2y/dx^2 + 5x (dy/dx)^2 - 6y = logx For each of the differential equations…
- (dy/dx)^3 - 4 (dy/dx)^2 + 7y = sinx For each of the differential equations…
- d^4y/dx^4 - sin (d^3y/dx^3) = 0 For each of the differential equations given…
- xy = a ex + b e-x + x^2 : x d^2y/dx^2 + 2 dy/dx - xy+x^2 - 2 = 0 For each of…
- y = ex (a cos x + b sin x) : d^2y/dx^2 - 2 dy/dx + 2y = 0 For each of the…
- y = x sin 3x : d^2y/dx^2 + 9y-6cos3x = 0 For each of the exercises given below,…
- x^2 = 2y^2 log y : (x^2 + y^2) dy/dx - xy = 0 For each of the exercises given…
- Form the differential equation representing the family of curves given by (x -…
- Prove that x^2 - y^2 = c (x^2 + y^2)^2 is the general solution of differential…
- Form the differential equation of the family of circles in the first quadrant…
- Find the general solution of the differential equation dy/dx + root 1-y^2/1-x^2…
- Show that the general solution of the differential equation dy/dx + y^2 +…
- Find the equation of the curve passing through the point (0 , pi /4) whose…
- Find the particular solution of the differential equation (1 + e2x) dy + (1 +…
- Solve the differential equation ye^x/y dx = (xe^x/y + y^2) dy (y not equal 0)…
- Find a particular solution of the differential equation (x - y) (dx + dy) = dx…
- Solve the differential equation [e^- 2 root x/root x - y/root x] dx/dy = 1 (x…
- Find a particular solution of the differential equation dy/dx + ycotx =…
- Find a particular solution of the differential equation (x+1) dy/dx = 2e^-y - 1…
- The population of a village increases continuously at the rate proportional to…
- The general solution of the differential equation ydx-xdx/y = 0 isA. xy = C B.…
- The general solution of a differential equation of the type dx/dy + p_1x = q_1…
- The general solution of the differential equation ex dy + (y ex + 2x) dx = 0…
- d^4y/dx^4 + sin (y^there there eξ sts there there eξ sts there there eξ sts) = 0…
- y′ + 5y = 0 Determine order and degree (if defined) of differential equations…
- (ds/dt)^4 + 3s d^2s/dt^2 = 0 Determine order and degree (if defined) of…
- (d^2y/dx^2)^2 + cos (dy/dx) = 0 Determine order and degree (if defined) of…
- d^2y/dx^2 = cos3x+sin3x Determine order and degree (if defined) of differential…
- (y″′)^2 + (y″)^3 + (y′)^4 + y^5 = 0 Determine order and degree (if defined) of…
- y″′ +2y” + y’ = 0 Determine order and degree (if defined) of differential…
- y′ + y = ex Determine order and degree (if defined) of differential equations…
- y″ + (y′) + 2y = 0 Determine order and degree (if defined) of differential…
- y″ + 2y′ + sin y = 0 Determine order and degree (if defined) of differential…
- The degree of the differential equation (d^2y/dx^2)^3 + (dy/dx)^2 + sin (dy/dx)…
- The order of the differential equation 2x^2 d^2y/dx^2 - 3 dy/dx + y = 0 isA. 2…
- y = ex + 1 : y′′ - y′ = 0 In each of the question verify that the given…
- y = x^2 + 2x + C : y′ - 2x - 2 = 0 In each of the question verify that the given…
- y = cos x + C : y′ + sin x = 0 In each of the question verify that the given…
- y = root 1+x^2 : y^there there eξ sts = xy/1+x^2 In each of the question verify…
- y = Ax : xy′ = y (x ≠ 0) In each of the question verify that the given functions…
- y = x sin x : xy′ = y + x root x^2 - y^2 (x ≠ 0 and x y or x - y) In each of the…
- xy = log y + C : y^there there eξ sts = y^2/1-xy (xy not equal 1) In each of the…
- y - cos y = x : (y sin y + cos y + x)y′ = y In each of the question verify that…
- x + y = tan-1y : y^2 y′ + y^2 + 1 = 0 In each of the question verify that the…
- y = root a^2 - x^2 x in (- a , a) :x+y dy/dx = 0 (y not equal 0) In each of the…
- The number of arbitrary constants in the general solution of a differential…
- The number of arbitrary constants in the particular solution of a differential…
- x/a + y/b = 1 In each of the question, form a differential equation representing…
- y^2 = a(b^2 - x^2) In each of the question, form a differential equation…
- y = a e3x + b e-2x In each of the question, form a differential equation…
- y = e2x (a + bx) In each of the question, form a differential equation…
- y = ex (a cos x + b sin x) In each of the question, form a differential equation…
- Form the differential equation of the family of circles touching the y-axis at…
- Form the differential equation of the family of parabolas having vertex at…
- Form the differential equation of the family of ellipses having foci on y-axis…
- Form the differential equation of the family of hyperbolas having foci on x-axis…
- Form the differential equation of the family of circles having centre on y-axis…
- Which of the following differential equations has y = c1ex + c2e-x as the…
- Which of the following differential equations has y = x as one of its…
- dy/dx = 1-cosx/1+cosx For each of the differential equations in question, find…
- dy/dx = root 4-y^2 (-2y2) For each of the differential equations in question,…
- dy/dx + y = 1 (y not equal 1) For each of the differential equations in…
- sec^2 x tan y dx + sec^2 y tan x dy For each of the differential equations in…
- (ex + e-x)dy - (ex - e-x)dx = 0 For each of the differential equations in…
- dy/dx = (1+x^2) (1+y^2) For each of the differential equations in question, find…
- y log y dx - x dy = 0 For each of the differential equations in question, find…
- x^5 dy/dx = - y^5 For each of the differential equations in question, find the…
- dy/dx = sin^-1x For each of the differential equations in question, find the…
- extan y dx + 1(1 - ex)sec^2 y dy = 0 For each of the differential equations in…
- (x^3 + x^2 + x + 1) dy/dx = 2x^2 + x, y = 1 when x = 0 For each of the…
- x(x^2 - 1) dy/dx = 1; y = 0 when x = 0 For each of the differential equations…
- cos dy/dx = a; y = 2 when x = 0 For each of the differential equations in…
- dy/dx = y tan x; y = 1 when x = 0 For each of the differential equations in…
- Find the equation of a curve passing through the point (0, 0) and whose…
- For the differential equation xy dy/dx = (x+2) (y+2) find the solution curve…
- Find the equation of a curve passing through the point (0, -2) given that at…
- At any point (x, y) of a curve, the slope of the tangent is twice the slope of…
- The volume of spherical balloon being inflated changes at a constant rate. If…
- In a bank, principal increases continuously at the rate of r% per year. Find…
- In a bank, principal increases continuously at the rate of 5% per year. An…
- In a culture, the bacteria count is 1,00,000. The number is increased by 10% in…
- The general solution of the differential equation dy/dx = e^x+y isA. ex + e-y =…
- (x^2 + xy)dy = (x^2 + y^2)dx In each of the question, show that the given…
- y^there there eξ sts = x+y/x In each of the question, show that the given…
- (x - y)dy - (x + y)dx = 0 In each of the question, show that the given…
- (x^2 - y^2)dx + 2xy dy = 0 In each of the question, show that the given…
- x^2 dy/dx = x^2 - 2y^2 + xy In each of the question, show that the given…
- xdy-ydx = root x^2 + y^2 dx In each of the question, show that the given…
- xcos (y/x) + ysin (y/x) ydx = ysin (y/x) - xcos (y/x) xdy In each of the…
- x dy/dx - y+xsin (y/x) = 0 In each of the question, show that the given…
- ydx+xlog (y/x) dy-2xdy = 0 In each of the question, show that the given…
- (1+e^x/y) dx+e^x/y (1 - x/y) dy = 0 In each of the question, show that the…
- (x + y)dy + (x - y) dx = 0; y = 1 when x = 1 For each of the differential…
- x^2 dy + (xy + y^2)dx = 0; y = 1 when x = 1 For each of the differential…
- [xsin^2 (y/x) - y]dx+xdy = 0 y = pi /4 whenx = 1 For each of the differential…
- dy/dx - y/x + cosec (y/x) = 0 y = 0 when x = 1 For each of the differential…
- 2xy+y^2 - 2x^2 dy/dx = 0 y = 2whenx = 1 For each of the differential equations…
- A homogeneous differential equation of the from dx/dy = h (x/y) can be solved…
- A homogeneous differential equation of the from dx/dy = h (x/y) can be solved…
- dy/dx + 2y = sinx For each of the differential equations given in question, find…
- dy/dx + 3y = e^-2x For each of the differential equations given in question,…
- dy/dx + y/x = x^2 For each of the differential equations given in question, find…
- dy/dx + (secx) y = tanx (0 less than equal to x pi /2) For each of the…
- cos^2x dy/dx + y = tanx (0 less than equal to x pi /2) For each of the…
- x dy/dx + 2y = x^2logx For each of the differential equations given in question,…
- xlogx dy/dx + y = 2/x logx For each of the differential equations given in…
- (1 + x^2)dy + 2xy dx = cot x dx (x ≠ 0) For each of the differential equations…
- x dy/dx + y-x+xycotx = 0 (x not equal 0) For each of the differential equations…
- (x+y) dy/dx = 1 For each of the differential equations given in question, find…
- y dx + (x - y^2)dy = 0 For each of the differential equations given in…
- (x+3y^2) dy/dx = y (y0) For each of the differential equations given in…
- dy/dx + 2ytanx = sinxy = 0 x = pi /3 For each of the differential equations…
- (1+x^2) dy/dx + 2xy = 1/1+x^2 y = 0 x = 1 For each of the differential…
- dy/dx - 3ycotx = sin2xy = 2 x = pi /2 For each of the differential equations…
- Find the equation of a curve passing through the origin given that the slope of…
- Find the equation of a curve passing through the point (0, 2) given that the…
- The Integrating Factor of the differential equation x dy/dx - y = 2x^2 isA. e-x…
- The Integrating Factor of the differential equation (1-y^2) dx/dy + yx = ay…
- d^2y/dx^2 + 5x (dy/dx)^2 - 6y = logx For each of the differential equations…
- (dy/dx)^3 - 4 (dy/dx)^2 + 7y = sinx For each of the differential equations…
- d^4y/dx^4 - sin (d^3y/dx^3) = 0 For each of the differential equations given…
- xy = a ex + b e-x + x^2 : x d^2y/dx^2 + 2 dy/dx - xy+x^2 - 2 = 0 For each of…
- y = ex (a cos x + b sin x) : d^2y/dx^2 - 2 dy/dx + 2y = 0 For each of the…
- y = x sin 3x : d^2y/dx^2 + 9y-6cos3x = 0 For each of the exercises given below,…
- x^2 = 2y^2 log y : (x^2 + y^2) dy/dx - xy = 0 For each of the exercises given…
- Form the differential equation representing the family of curves given by (x -…
- Prove that x^2 - y^2 = c (x^2 + y^2)^2 is the general solution of differential…
- Form the differential equation of the family of circles in the first quadrant…
- Find the general solution of the differential equation dy/dx + root 1-y^2/1-x^2…
- Show that the general solution of the differential equation dy/dx + y^2 +…
- Find the equation of the curve passing through the point (0 , pi /4) whose…
- Find the particular solution of the differential equation (1 + e2x) dy + (1 +…
- Solve the differential equation ye^x/y dx = (xe^x/y + y^2) dy (y not equal 0)…
- Find a particular solution of the differential equation (x - y) (dx + dy) = dx…
- Solve the differential equation [e^- 2 root x/root x - y/root x] dx/dy = 1 (x…
- Find a particular solution of the differential equation dy/dx + ycotx =…
- Find a particular solution of the differential equation (x+1) dy/dx = 2e^-y - 1…
- The population of a village increases continuously at the rate proportional to…
- The general solution of the differential equation ydx-xdx/y = 0 isA. xy = C B.…
- The general solution of a differential equation of the type dx/dy + p_1x = q_1…
- The general solution of the differential equation ex dy + (y ex + 2x) dx = 0…
Exercise 9.1
Question 1.Determine order and degree (if defined) of differential equations given
![](data:image/png;base64,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)
Answer:It is given that equation is ![](data:image/png;base64,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)
⇒ y”” + sin(y’’’) = 0
We can see that the highest order derivative present in the differential is y””.
Thus, its order is four. The given differential equation is not a polynomial equation in its derivative.
Therefore, its degree is not defined.
Question 2.Determine order and degree (if defined) of differential equations given
y′ + 5y = 0
Answer:It is given that equation is y’ + 5y = 0
We can see that the highest order derivative present in the differential is y’.
Thus, its order is one. It is polynomial equation in y’. The highest power raised to y’ is 1.
Therefore, its degree is one.
Question 3.Determine order and degree (if defined) of differential equations given
![](data:image/png;base64,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)
Answer:It is given that equation is ![](data:image/png;base64,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)
We can see that the highest order derivative present in the given differential equation is![](data:image/png;base64,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)
Thus, its order is two. It is polynomial equation in
and ![](data:image/png;base64,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)
Therefore, its degree is one.
Question 4.Determine order and degree (if defined) of differential equations given
![](data:image/png;base64,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)
Answer:It is given that equation is ![](data:image/png;base64,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)
We can see that the highest order derivative present in the given differential equation is
.
Thus, its order is two. The given differential equation is not a polynomial equation in its derivative.
Therefore, its degree is not defined.
Question 5.Determine order and degree (if defined) of differential equations given
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALcAAAA7CAYAAAA5HBs8AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAVgSURBVHja7VwxbtswFOUBcoAWMLP1ADGdMVvhCEUOkETw1KJTQaDoXCjZAhS5QLZsydC5Q7t2yQ0y5Aa5Q5svmzRFUzZpi1QkveEBtVLb4ufT5+Pj92eXl5cMAPoIBAEAuQEA5AaAPpH7IhcnjLF/JcT5LQIL9ILchcwOuDi/1v9m7JlPv3xDcIHOk1vOZkefr6721OsvU/6NjbI/5jUA6IXmLiUKyA30kdyUuSFLgE6RW+bZhzFnD+uIWxSzt2L/+AZZG+gMuUsdvXBD1pE7F6KYFcVbBBboVOZWdl8duekBEPnFCYIK9Irc9DfzOr0G0YHOk9uULEuIJ8gT4NWSu5Bn72gDWZKVjx9mM/Fpk+Z2bjAZe9KkX1iEV1ef97IR+2New2QAScitSMmzvCDimUQPtfnovXOCV7M5fceYjX+fyeIdJgJIRu5zwW7tjKrkxzYetnqvqcPhhwPJyV1XH7LJLfGSJ4uiqlKWcH6fyeIAkwAkI3cdiXcht14NGH8mQpcPEM/uobWBXpDbXBEgSYBWyW3XZe9Kbu2Q8PHDZMLvIEmA9Jpb23dVd6OO9CHQnwH7D2iD3Ka7oUhIdt6x2L+x/eptszckCdAauQlyKj6RRiYy74vjG5kfntKG8DDLv+6Sdc8Fv4Yk6TZoDpU50Elyx8DQS2Irp76Lk98uHmI1Se6YMUkalCG7JOQWCZH9UA/2clKHW4cTOyZJCD30giraa0zF9Lu9Yil7dIgVlCliEn0QZTksafeOLsGxM1dscndttewUuV+r9p1w/kutKLRhtjMIZZY8O/yqNtWkMamFhev/aSeJqifl7EiIvPDTruxWWasrFZSLM4Xq9XCdG0Jun7Eo54wz/qjuRceKs0cipfpJYjkGR8xSxWSw2fIwl6fma9Pe1AdOxrW5W7Rqg9JkKPLQ+8zJqYOUZwfl5E+yn+Zn6fdb3zM/HxBP26x8IeTeNJZSIztIpe95kSiyXH4wJalPFo4Rk86Q0v2jCDfqJlOT1iLffPe/3A/Mv2s1I9iVkSqDmJNXXqvJ3JU6dnWvVmYz/w997q5FZr7k9h3L8v6q8XER2adJU8yYQOPWBdtxUGVn+cpEL7KV7/JPK4GSPPbkm6fEVKqwi/70JXfIWMwiOB9y+5xqx4hJ9YVnZoyJ2JvbTcukDqKL3I7SBC1XSp3KH5XcCXnYXC7SNqUKIaubKwa+Y4lB7hgxGVTm9in+WkfuZXazflVklycETOacKO6JHI1Gf3cpVtvGLfEZS0xyNxmTQWnudVlhk6asEL9msszTNl8p4eqtqOrd5YV8U3cvsa3AdWOJTe6mYjK4gwO9eXkJuBk8sx1F3U/qbM3uOogI8WldhWT0AE345E49fD4PZFOa23csMcndZExakwZt9fKufL8Gd1uBL39TutPltOiJMHb380lelS1lxi8PsuTB0uLi16r9c93EVlatl3sMtQNDN5Qbx2J4zCaRlV1n96+p08gpYpLerXgFvbzpu81iHfJm7QCpQJsPgF0RWQY+yz9K+f7IbIPh+qyKjiUS8Mkv25Uw/WIVl1U5FiZRgsi9YSzLjGmMI8uLFStv8Wuryn1bBE8Rk6SkQi9vICXady9AbqBtcvu0ME61kweAxsjt28I4SPeilzfwWjL3rr9+t4Fe3kAvyY1e3kAvyY1e3kDr5F7XwnjbInL08gZaJ7dPC+MYhfUAEJ3cvi2Mmy6sB4Co5A5tYdxkYT0ARCX3Nl1e0QMQ6DW5myisB4Ak5PZtYdxkYT0AxNXcAS2Mmy6sB4Co5Cb4tDBWmbrJwnoAiE5uwqYWxjEK6wEgCbkBAOQGAJAbAEBuAGgE/wE27YQKubBmPwAAAABJRU5ErkJggg==)
Answer:It is given that equation is ![](data:image/png;base64,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)
![](data:image/png;base64,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)
We can see that the highest order derivative present in the given differential equation is![](data:image/png;base64,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)
Thus, its order is two. It is polynomial equation in
and the power is 1.
Therefore, its degree is one.
Question 6.Determine order and degree (if defined) of differential equations given
(y″′)2 + (y″)3 + (y′)4 + y5 = 0
Answer:It is given that equation is (y”’)2 + (y”)3 +(y’)4 + y5 = 0
We can see that the highest order derivative present in the differential is y”’.
Thus, its order is three. It is polynomial equation in y”’, y” and y’
So, the highest power raised to y”’ is 2.
Therefore, its degree is two.
Question 7.Determine order and degree (if defined) of differential equations given
y″′ +2y” + y’ = 0
Answer:It is given that equation is y″′ +2y” + y’ = 0
We can see that the highest order derivative present in the differential is y”’.
Thus, its order is three. It is polynomial equation in y”’, y” and y’
So, the highest power raised to y”’ is 1.
Therefore, its degree is one.
Question 8.Determine order and degree (if defined) of differential equations given
y′ + y = ex
Answer:It is given that equation is y’ + y = ex
⇒ y’ + y – ex = 0
We can see that the highest order derivative present in the differential is y’.
Thus, its order is one. It is polynomial equation in y’
So, the highest power raised to y’ is 1.
Therefore, its degree is one.
Question 9.Determine order and degree (if defined) of differential equations given
y″ + (y′) + 2y = 0
Answer:It is given that equation is y’’ + (y’)2 + 2y = 0
We can see that the highest order derivative present in the differential is y”.
Thus, its order is two. It is polynomial equation in y” and y’
So, the highest power raised to y” is 1.
Therefore, its degree is one.
Question 10.Determine order and degree (if defined) of differential equations given
y″ + 2y′ + sin y = 0
Answer:It is given that equation is y’’ + 2y’ + siny = 0
We can see that the highest order derivative present in the differential is y”.
Thus, its order is two. It is polynomial equation in y” and y’
So, the highest power raised to y” is 1.
Therefore, its degree is one.
Question 11.The degree of the differential equation
is
A. 3
B. 2
C. 1
D. not defined
Answer:It is given that equation is ![](data:image/png;base64,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)
We can see that the highest order derivative present in the given differential equation is
.
Thus, its order is three.
The given differential equation is not a polynomial equation in its derivative.
Therefore, its degree is not defined.
Question 12.The order of the differential equation
is
A. 2
B. 1
C. 0
D. not defined
Answer:It is given that equation is ![](data:image/png;base64,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)
We can see that the highest order derivative present in the given differential equation is
.
Thus, its order is two.
Determine order and degree (if defined) of differential equations given
Answer:
It is given that equation is
⇒ y”” + sin(y’’’) = 0
We can see that the highest order derivative present in the differential is y””.
Thus, its order is four. The given differential equation is not a polynomial equation in its derivative.
Therefore, its degree is not defined.
Question 2.
Determine order and degree (if defined) of differential equations given
y′ + 5y = 0
Answer:
It is given that equation is y’ + 5y = 0
We can see that the highest order derivative present in the differential is y’.
Thus, its order is one. It is polynomial equation in y’. The highest power raised to y’ is 1.
Therefore, its degree is one.
Question 3.
Determine order and degree (if defined) of differential equations given
Answer:
It is given that equation is
We can see that the highest order derivative present in the given differential equation is
Thus, its order is two. It is polynomial equation in and
Therefore, its degree is one.
Question 4.
Determine order and degree (if defined) of differential equations given
Answer:
It is given that equation is
We can see that the highest order derivative present in the given differential equation is.
Thus, its order is two. The given differential equation is not a polynomial equation in its derivative.
Therefore, its degree is not defined.
Question 5.
Determine order and degree (if defined) of differential equations given
Answer:
It is given that equation is
We can see that the highest order derivative present in the given differential equation is
Thus, its order is two. It is polynomial equation in and the power is 1.
Therefore, its degree is one.
Question 6.
Determine order and degree (if defined) of differential equations given
(y″′)2 + (y″)3 + (y′)4 + y5 = 0
Answer:
It is given that equation is (y”’)2 + (y”)3 +(y’)4 + y5 = 0
We can see that the highest order derivative present in the differential is y”’.
Thus, its order is three. It is polynomial equation in y”’, y” and y’
So, the highest power raised to y”’ is 2.
Therefore, its degree is two.
Question 7.
Determine order and degree (if defined) of differential equations given
y″′ +2y” + y’ = 0
Answer:
It is given that equation is y″′ +2y” + y’ = 0
We can see that the highest order derivative present in the differential is y”’.
Thus, its order is three. It is polynomial equation in y”’, y” and y’
So, the highest power raised to y”’ is 1.
Therefore, its degree is one.
Question 8.
Determine order and degree (if defined) of differential equations given
y′ + y = ex
Answer:
It is given that equation is y’ + y = ex
⇒ y’ + y – ex = 0
We can see that the highest order derivative present in the differential is y’.
Thus, its order is one. It is polynomial equation in y’
So, the highest power raised to y’ is 1.
Therefore, its degree is one.
Question 9.
Determine order and degree (if defined) of differential equations given
y″ + (y′) + 2y = 0
Answer:
It is given that equation is y’’ + (y’)2 + 2y = 0
We can see that the highest order derivative present in the differential is y”.
Thus, its order is two. It is polynomial equation in y” and y’
So, the highest power raised to y” is 1.
Therefore, its degree is one.
Question 10.
Determine order and degree (if defined) of differential equations given
y″ + 2y′ + sin y = 0
Answer:
It is given that equation is y’’ + 2y’ + siny = 0
We can see that the highest order derivative present in the differential is y”.
Thus, its order is two. It is polynomial equation in y” and y’
So, the highest power raised to y” is 1.
Therefore, its degree is one.
Question 11.
The degree of the differential equation is
A. 3
B. 2
C. 1
D. not defined
Answer:
It is given that equation is
We can see that the highest order derivative present in the given differential equation is.
Thus, its order is three.
The given differential equation is not a polynomial equation in its derivative.
Therefore, its degree is not defined.
Question 12.
The order of the differential equation is
A. 2
B. 1
C. 0
D. not defined
Answer:
It is given that equation is
We can see that the highest order derivative present in the given differential equation is .
Thus, its order is two.
Exercise 9.2
Question 1.In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
y = ex + 1 : y′′ – y′ = 0
Answer:It is given that y = ex + 1
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGAAAAArCAMAAACw2X2lAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZjoAZjo6ZjpmZmYAZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkLbbkLb/kNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29uQ2////7Zm/7aQ/9uQ/9u2//+2///bENfl+AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACM0lEQVRYR91Xa1PCMBBMqkJ8QFUUFSJKfRSVWpv//+O8tE2btPaSg3HGMR8cgeW299osjP2Pk3G+JGRChOvIGYmACgeCXFAyoMJ/myC/4nwilkpyPtrmgkdrtB01PLxluVgwtYIeKDnaQjd88Q08mCEpw+ompwev8GeMf7OFBzIUs0vTA91pdY8XyILTCZQcs0zng5y9CHSNEs+40gn0Y5tFK2bnc2gDdmx4YI3S6IF9XVebnHDdEPTYcB+2+lw98Sj+ECVDpufIk4IF92H7n2eeGR2IqJ7P+g+WTl9cOAyob0aH4ssLa/TeRLTOBQxjfuoOjJLxaqcElHQbB5OexpoxP3Z2CrRoiu/AQAK9xqWHdxU09SxVWA+rwbXP5qQO3EkhLF4P1VwlILORfvTPRyWPygZ3i7cbg5FfrcrvoGYbMVYJCL+OlmBNTbg5uIAbAi2z3XLp95ow7T9DmfwMLUWesUqkSjFvT+flXiUqZiW/KwUoAbFEVQadg/YgNJ96ivrTasoWGmgIZyJnHO72Tb1iFbjdA4JV60PNJm/AkNw6PbY2meDsetDBdbIXmeDs+lDlqGlTSkdN9yJgnvvAWDWYSxBa1GcToO1MNM6OJaAJSiK2ggDtrnPZOe0dU+zWb12dF9oS2EYnF5MYWQsC1IriOKkUFU4CdICgmN9gvsUm8EAtAsuqgakoZsgVQYDahW6smkqgwRnm7QhQO4Xaqi0k/LbRqj48R8YEBkD3ldA/9P1vP01CglggyYUAAAAASUVORK5CYII=)
Now, Again, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
⇒ y” = ex
Now, Substituting the values of y’ and y” in the given differential equations, we get,
y” – y’ = ex - ex = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question 2.In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
y = x2 + 2x + C : y′ – 2x – 2 = 0
Answer:It is given that y = x2 + 2x + C
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
⇒ y’ = 2x + 2
Now, Substituting the values of y’ in the given differential equations, we get,
y’ – 2x -2 = 2x + 2 – 2x - 2 = 0 = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question 3.In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
y = cos x + C : y′ + sin x = 0
Answer:It is given that y = cosx + C
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
⇒ y’ = -sinx
Now, Substituting the values of y’ in the given differential equations, we get,
y’ + sinx = -sinx + sinx = 0 = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question 4.In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
![](data:image/png;base64,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)
Answer:It is given that y = ![](data:image/png;base64,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)
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOsAAAAvCAMAAAA1rc1FAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmYAZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29uQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///bN1vkEgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADkElEQVRoQ+1Za1vTMBROho6qsClVQJ1FLkWMQ6Cdzf//Z54kvSRp1iax6dbHnW97SE/O/bxvQOgghwiMGoEM42+jXjjEZY8Xd15qssn5ujnHRz+9fM2jieU1XzyQ/8VXyKiPr/k5xqdTy6ufr3m0QvR6cv3qldd0/oLQ9GaTj69FfAb1MLnZ5FXDB1+9FtfIH7nPYZocg42T7NeZM24is1u0uZjaHKY3Ecb41Ve3WqL3eLb8Hf2rs0+wpZduN6MiZjtg10J/nLiBzeLjLVpDUZHFg73tezH9afLBI+D5G2ig/J09ss48Ea19NPtP0oTt3EqsuRPhX3GP7YSwsbhjkeOtcyeabMvbumxXYmjCX6aPtqsaz3+xhoS0uNNWA0k1mUyJbTqTJhjPX/IIz+6Kzx6NMnAYtJGh7mzF15TZTfijRgauZjx7ageUtpF6GdKEA3D35Tiwl0KdZkeHrygFk7n3RQzrrtxbKZSF+C1L7RzXp7UqBK2UkWPg4CtLkhhJknDmpAtNqqHLyoZe9Q0wPVSev2szDN+zv2lAs/lpiH4enbYQRI+vbBxk+wAiHGsYAtMuOuZrRw2zWKbaXJ5CDaPi8ksLELB+1UUOSRG/v3RDZUHGktgz6uNT12yCvitizTXBiFVRVaa4fcLGHZj4rsKmZBdikfcra1+lSpWdQ1MwOtMsN+1XFUt4YsP+gWYKmOyr4c1fsqXFnWRfARcAIoDtoqTJhJtUIzI/bDgA9mjzdCMasCky3gHdeBiS45cfKCCXEJk5poFZ+fEc0ew9PIcmy2srm+kjMOLX8houh7cVF6k4ZufcEH905q+Vzl7+CnW/sILBKV6hTSwNefqdDW97LqJXWPXmLwFb21oNfY5vLiI9l+QnECN7LgIfK2OkefOvgW1oF9z0y7OkMt2OiyBUcczyxubN3whs3ewKcTrlNfznmem+L3vXiotAVlXsKr+Dm4BtCOuddPKxubnBrJyLTyXW6sQ2NRdpOKa4UXnzNwBbJ7sCHBa77+mZZ7feOFZcROaYLV9NwDaA+S4qa/yWR+AzqehCV16NXERslubN3wRsXewKcTY9LpcT474NAOn0dXvK6jd/I7ANYb+DTo41M743YBzzjcPFlYuUn1Vv/isjsHWwK8BRjjV5DlgBztf1snTlIgFsG1plSd+FiwS/Fe3qwUWGNiy0vkIGi6Ev27X+ffgXwa5jcLh/1Aj8BUEoaeBnoLMAAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, LHS = RHS
Therefore, the given function is the solution of the corresponding differential equation.
Question 5.In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
y = Ax : xy′ = y (x ≠ 0)
Answer:It is given that y = Ax
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
⇒ y’ = A
Now, Substituting the values of y’ in the given differential equations, we get,
xy’ = x.A = Ax = y = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question 6.In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
y = x sin x : xy′ = y + x
(x ≠ 0 and x > y or x < – y)
Answer:It is given that y = xsinx
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHMAAAArCAMAAAB8QEdOAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpC8OpDbZgAAZjoAZjo6ZjpmZjqQZmYAZrbbZrb/kDoAkDo6kGYAkLbbkNv/tmYAtmY6tpBmttvbttv/tv//25A625Bm27Zm27aQ29uQ2////7Zm/9uQ/9u2//+2///b8qcStwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACQ0lEQVRYR+1W11bDMAy1CiWsUkZLGW5pWGY0w///c0hKnDhpQpyUYx7ADzk9laWreS0h/k9TBiKAe++ZiX4BMw78x+kbM54DTPzGGQd3Qq/81jMcb4Tw20Pp7AqnxG89/wqmlgeYW7/1FGr0IJJrv32rH2E0/Qx+FDSd0TD0PPrp9NVZRZ291O4OGQMtL/r4GR/X2Drac/c4d1dLmtnt09pp8eG6cltRX/Y7bW62d7eq1E/L3q9UNj69DgUqAcabOIDROr3pUxkGyjsgJBuKNohkCdjbG94m6PMWwFSIUi4EVUNLZu5RNc2Orhu1ENUpTWRMr9AgOxMHJ2uhaLSMnMyGmBpFnVMrJTqWn+99MZiEpbCbmJLj0wKTgfEvI2fMcZYfvewIs/Ch/EH67G+W5AnmUIhHOHkuks7Bmpchk+eY1AfRADIg/bIkKk/I+xHsox95bgtM5E0TFcVJzoa1ju2bW5Eubk3IHwEWrI5pyameGP35ojcZmJRmzmJt0hlBYSmxYHY9ObdGzmjMIiE0k0l38+bzqUM0EKEV2piSOYIzJ/CHeqiQU+UzInLlvIYdnlW1pOmeAVwlywDgjMLE2eQP/X1ZyKmSWetErmSyzWhtfNuaIg5zue4clMJAw8Mz6F2R05VrmI0735D3U1L+HY7Z4W3udFDb5Uq5w9vcuYvFbt1yh7e5s1tvhxv2Pm24dQdzTqqVHb7kTifdoZdsTIs7h5pz0rN2eIs7nVSHXyp2eJs7h5tz0jQ7/J3NnU6af/fSF5ADP89VRQdbAAAAAElFTkSuQmCC)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPkAAAArCAMAAACNZ96GAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpC8OpDbZgAAZjoAZjo6ZjpmZjqQZmYAZrbbZrb/kDoAkDo6kGYAkLbbkNv/tmYAtmY6tpBmttvbttv/tv//25A625Bm27Zm27aQ29uQ2////7Zm/9uQ/9u2//+2///b8qcStwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADmElEQVRoQ+1Za5eTMBBNqhV1H/XRdbXSWtYHqxZK/v+fc5ISGAJJBhg+eE750LPtTu7MnWQmN0SI63PNwIIZKKT8ygvPj8gbX4NWcDMX/IjLUC8T5jkX/IhX5nwZKB+kvGOdc35EPrYIqUx2Qh0465wfcRHiIlufBG8/4kdchHm1/Qi4nP2IH3ER4oI/Tn7EK3PeDKj0FQBy6g5+RF7GDVq++ibOnzh7Oz/iMtTVk1xt/iaM1PkRB5hXW70n8T/q++2vdlXc/5zvoYPogctHOOLcjVA4Kn2PE1q+na3qHUQP9RGOihft1HjnZXT7Uqne47HOe32cN+sOojeikuwo14059oxm3s9njotKpaOXgIPoj6jj6ELsecjbhBhiaYL/X7Yl/6SP9zqA6Fvv/UlvK1qlUq5PZSJXx+pzqL+d9xLa9sm8HtEfz4ncCJHp0XnoJUztCht2VitmHkfTFG3w8Yh6hQaj85WtNZWaQ0bz3ZM+baYOYGr8lsnNUeR678pgYHDaLDI2zNAy6AyOounoakRKROCo2krnaajmuq85JQ7Jrx9rZvRzedswN/ThJ+0/d1pYJ3mWOTY056/66TCPoiHmlIiydX8iVWobuZ5EtY+22yd586NZa2bi7WHiDlY9PG5q4bv+2ST2MrA2hIVSM+8nGBlFEaMRaUch5rphFAQJ8/uNfAkk6tXeMEeFM1gobR21Feadc1yGnrJDlRmPCJgHVruelczp9f3J0HH8SaA+XebV45egDGiYI0NvnYsYGlrtlIiwozqTbf71qn33GJUwusTNKnWYQ51U25AOsJ0YGV7KZLDOY2hNvV2aTiSijiNbc3iSMxlqUXbETpwfgKLRDeZDdziVwdAiNL7efbFhR1vhDtcYBZNZI+oXdJGIBkRcV8lQVOt5n0h5r6ccNm/zoQvoQ6p1APxhUzdwG2LQQTa0hj4N1xoh5j5EQYhoQMN1u0dBUa3ehuNg9Q6oc3V7X5wOCZSh8MK6HcqPsKURaZu9qyeMZ57V4oie6CJnNZVuDoxTPvjedd75fOjszHE+h9qC6uV57G0ITX5TfPIjUryOtmlvQ0jym4DPj0hwOsGkvQ0hyW+CB35EgtPxJvhOoCO/x0PVI/gRJ4cSHNi5DcEacbI7fsTJoZCZU+R3PArMnAcx7nOKBboNiYp5Gj4/Is3vaKvmNiQu5onY/IhExyPN7G3IzhXzI3Fac37EyaFcB/4/GfgHYg+AHeObTvwAAAAASUVORK5CYII=)
⇒ y’ = sinx + xcosx
Now, Substituting the values of y’ in the given differential equations, we get,
LHS = xy’ = x(sinx + xcosx)
= xsinx + x2cosx
= y + ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= RHS
Therefore, the given function is the solution of the corresponding differential equation.
Question 7.In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
xy = log y + C : ![](data:image/png;base64,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)
Answer:It is given that xy = log y + C
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ y + xy’ = ![](data:image/png;base64,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)
⇒ y2 + xyy’ = y’
⇒ (xy – 1)y’ = -y2
⇒ y’ = ![](data:image/png;base64,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)
Thus, LHS = RHS
Therefore, the given function is the solution of the corresponding differential equation.
Question 8.In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
y – cos y = x : (y sin y + cos y + x)y′ = y
Answer:It is given that y – cosy = x
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ y’(1 + siny) = 1
⇒ y’ = ![](data:image/png;base64,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)
Now, Substituting the values of y’ in the given differential equations, we get,
LHS = (ysiny + cosy + x)y’
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= y = RHS
Therefore, the given function is the solution of the corresponding differential equation.
Question 9.In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
x + y = tan–1y : y2y′ + y2 + 1 = 0
Answer:It is given that x + y = tan-1y
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAL4AAAArCAMAAADxAcJfAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmZmOmaQOma2OpCQOpC2OpDbZgAAZjoAZjo6ZjpmZmYAZmY6ZrbbZrb/kDoAkDo6kGYAkLyQkLbbkNv/tmYAtmY6tmZmtpBmttv/tv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/9uQ/9u2//+2///bnSzzjwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADL0lEQVRoQ+1Y21bbMBCUAq17o6TFQEsMNE5bTC+xsf7/37qSJUuytLJIvRwe0ENOTmLvjEej3fUy9rKegwIt5xtiHqQQLTl9RgnRFdTqM0oIytjalVQQ3TnnJ7TqE0J0xRUTN6Tep4Soj/eM9FwxRgjRl2dgTipjKt9TQlDG1oeWEoIy9hPQF9VrQKGsKYwUoll9Yw8XpJmHEkLs+Gr9p6Dk/18QYift8eyW+PHhLoPU/cfzKf3m9OfMjZmxIcp8rDiUqD5DvclZTaB+9y7doeXHhpIxEwtjL8tN1grps+7NNnGrqKKxRRV96HQsBKY9Us7B7u1LrpYEjNBnjWwVsKVjT/9G6KdjYeIPjmhXKRmHe2P0U5INeZrVnB/vG+c9zKEvKvlnV3AJf4D8uknZSYWP7sSvt5yvZfXhm/tCfnOWqF8ppT1IxB/qNtMA1cDNVdz/rjo9JV4qFuYdrboKAHLthQQD5Pdb1ng5HPTjXHlZVBaS1Xg2NTsqr2+cU+CZp5Hm1ftqY8GO6ZV2hYGw5lGa6Q/kVHuQ8Cgjlv3iGbIrTsxGBrwkkrgenKsa4vhCIBQTz/uW/tCFRVYmpIkN6noaeurLA9Jq2gn6OeZh4vdFoc6YegaUfibkuKP95RetkmLhZx54yFonUkv/IPNI7xvfpOgzFzLD+9fbvnQu8+n35adLXfUT5whR32QHpZT6yDAP7IuBRHfIZh5Rgwfb4dRH1IfEqv9KxULo69wMpM/Y91v4ePgqfaP6dvkbskZIm6vD8dcQG/LsCtTXWSs0DxuL22zej0zYzM3Q7m4YpP/135JvoI7wjQc5eYwR0qm64asKUnWnsYyvkhV8SDBBO35ArZBxNKQrWPiaPh8bUqbJmjlFNzIJeExXOFjXQnpd4iGxRbW+MUpkdJyxQUZ+T25OnoG0PboZf037m7nYcDBOh1I12+9jENjxxH63kOMVdvw17W8eGxy9nhLCjr+m/c1i9Akh3BmP098sRh0CUUJ4Iyq/v1nqESgh3NiT/oaC/tIQzvgLsqrX3yxFnxRiHH8F/c1S9KHn1kM8Aggz/roK+pvF6D8BxGJcXwLlK/AP+O1py6JEeAcAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAI0AAAAyCAMAAABWHKg3AAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOgA6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZpDbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///brJv9kwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACNklEQVRYR+2Z61KDMBCFE7XFVmu9FG9VrBYk7/+EJhBy3SR0SIjjmH8tofOx2d2zpyD0v+JEoN1gjK/Dv/W1xXitbKvobefv4ftO29FuFscRd7Q3L+hw9qruJGU2GobRXP4mmko/0LyxOahpQ0OVlaYyYGanISXGi2NTYJq+NYWpd1mzmJSs4GoK0zUCnJcGVayIqyXYAubImz2LQbdYd2mKHSKPWmELsjlo9DCQcolqR3ucn4Yd1V7LFkmbgabdXN051CgDDdo7JTUHTe2Uxpg0g3aHNLyGyzuyMrDaZctHQ2vbVd6RaYYT8NKU6ydnaII69XkL9ymolQ4N1k+DV+5RzJ8331t4NPyA+gUp+behvHHPhV6aZvXWqYq1hgxhE4nQ5PaeP3QiGkoB06BKzLNSkwVzcppe/dUlcGxNjkGjK+3woI7YKNlva3JPY+IHP4+o8DCNrckxYgNnOqfxnJStyTFoTjopmcWs9RqaHIPGGxvjoqxwdsHU5IQ0ujXlVHr3MzX5ZBphyL3djzwXtBIuHgKm2tRkSTNOV6QhnzpRQJo80Fi6IrQD6O6dIZ9MA2gyp7F1xUPTG/LpNLYmy5MyupVKo5pOhLghn0oDpdQoGhqHwXRSNeSGPB+NYjqFIZ+Nxm6uQuCkIZ+Npk9SdUoDTGdGGsB05qSxTWdiGvMvT32etkxnQhpbV8zuZ5nOhDQBdaOXLdOZjQY0nfloINOZkQYwnWloxr31MBMrzVuPcPr+pR0/AKlGfTi2Fd0AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
Now, Substituting the values of y’ in the given differential equations, we get,
LHS = y2y’ + y2 + 1 = ![](data:image/png;base64,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)
= -1 – y2 + y2 + 1
= 0 = RHS
Therefore, the given function is the solution of the corresponding differential equation.
Question 10.In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGQAAAAaCAMAAAB2FAnXAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OjoAOjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmYAZmY6Zrb/kDoAkDo6kDpmkGYAkGY6kLbbkNv/tmYAtmY6tv//25A627Zm27aQ29u229v/2/+22////7Zm/9uQ/9u2/9vb//+2///bQnjujwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABh0lEQVRIS+1Va1PCMBBMBCoqWh+oBB8UUsEmmPz/f+dd2vQRaNJ2Bh1nvE+d6d7d3u5dS8h//GUFOB0YPYbWy1UP9ECoeswGZvZIE5MaeHtP6Syc3BFWFUoW1bO6fSfpWVC+jrCqrn7ZNJnL82ATTOgIy2vLC8cSfhOWCxAIY5SOMxnR4Oympk6n1ou01ZIES8K+G3kNTLMxMBS+Hl87RK9BHM0mmU4MlntsR4RmpkcB4yPQmtvNARZF2Lb7N4ov1Z21REaQLqCHqG1CQzkknotpYZjju7PtLkEa1QJjgoqRSt7kkBeR0aUZtISBBESgZK0hI2DFTUX98RDZ4j7buSs/6FWdwBFaIO9ok8+KngDNNpnKtmr+hNPXQsXXc+cEHJKCLvIFNvsRbgKMVFz/PhhRAxsPRqYGgsLtn0OT6ATAwikqnNEOxeZ0mku0hiv5jAvHW0zRDI4OLW9Qb3z5jmaqOMjDtwZkuerynyjvyFur9SWbvToWDavjzWL06gf+RScg/islvwHlSB2PozWcqwAAAABJRU5ErkJggg==)
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, Substituting the values of y’ in the given differential equations, we get,
LHS = ![](data:image/png;base64,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)
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= x –x = 0 = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question 11.The number of arbitrary constants in the general solution of a differential equation of fourth order are:
A. 0
B. 2
C. 3
D. 4
Answer:As we know that the number of constant in the general solution of a differential equation of order n is equal to its order.
Thus, the number of constant in the general equation of fourth order differential equation is four.
Question 12.The number of arbitrary constants in the particular solution of a differential equation of third order are:
A. 3
B. 2
C. 1
D. 0
Answer:In a particular solution of a differential equation, there is no arbitrary constant.
In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
y = ex + 1 : y′′ – y′ = 0
Answer:
It is given that y = ex + 1
Now, differentiating both sides w.r.t. x, we get,
Now, Again, differentiating both sides w.r.t. x, we get,
⇒ y” = ex
Now, Substituting the values of y’ and y” in the given differential equations, we get,
y” – y’ = ex - ex = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question 2.
In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
y = x2 + 2x + C : y′ – 2x – 2 = 0
Answer:
It is given that y = x2 + 2x + C
Now, differentiating both sides w.r.t. x, we get,
⇒ y’ = 2x + 2
Now, Substituting the values of y’ in the given differential equations, we get,
y’ – 2x -2 = 2x + 2 – 2x - 2 = 0 = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question 3.
In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
y = cos x + C : y′ + sin x = 0
Answer:
It is given that y = cosx + C
Now, differentiating both sides w.r.t. x, we get,
⇒ y’ = -sinx
Now, Substituting the values of y’ in the given differential equations, we get,
y’ + sinx = -sinx + sinx = 0 = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question 4.
In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
Answer:
It is given that y =
Now, differentiating both sides w.r.t. x, we get,
Therefore, LHS = RHS
Therefore, the given function is the solution of the corresponding differential equation.
Question 5.
In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
y = Ax : xy′ = y (x ≠ 0)
Answer:
It is given that y = Ax
Now, differentiating both sides w.r.t. x, we get,
⇒ y’ = A
Now, Substituting the values of y’ in the given differential equations, we get,
xy’ = x.A = Ax = y = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question 6.
In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
y = x sin x : xy′ = y + x (x ≠ 0 and x > y or x < – y)
Answer:
It is given that y = xsinx
Now, differentiating both sides w.r.t. x, we get,
⇒ y’ = sinx + xcosx
Now, Substituting the values of y’ in the given differential equations, we get,
LHS = xy’ = x(sinx + xcosx)
= xsinx + x2cosx
= y +
= RHS
Therefore, the given function is the solution of the corresponding differential equation.
Question 7.
In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
xy = log y + C :
Answer:
It is given that xy = log y + C
Now, differentiating both sides w.r.t. x, we get,
⇒ y + xy’ =
⇒ y2 + xyy’ = y’
⇒ (xy – 1)y’ = -y2
⇒ y’ =
Thus, LHS = RHS
Therefore, the given function is the solution of the corresponding differential equation.
Question 8.
In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
y – cos y = x : (y sin y + cos y + x)y′ = y
Answer:
It is given that y – cosy = x
Now, differentiating both sides w.r.t. x, we get,
⇒ y’(1 + siny) = 1
⇒ y’ =
Now, Substituting the values of y’ in the given differential equations, we get,
LHS = (ysiny + cosy + x)y’
= y = RHS
Therefore, the given function is the solution of the corresponding differential equation.
Question 9.
In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
x + y = tan–1y : y2y′ + y2 + 1 = 0
Answer:
It is given that x + y = tan-1y
Now, differentiating both sides w.r.t. x, we get,
Now, Substituting the values of y’ in the given differential equations, we get,
LHS = y2y’ + y2 + 1 =
= -1 – y2 + y2 + 1
= 0 = RHS
Therefore, the given function is the solution of the corresponding differential equation.
Question 10.
In each of the question verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
Answer:
It is given that
Now, differentiating both sides w.r.t. x, we get,
Now, Substituting the values of y’ in the given differential equations, we get,
LHS =
= x –x = 0 = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question 11.
The number of arbitrary constants in the general solution of a differential equation of fourth order are:
A. 0
B. 2
C. 3
D. 4
Answer:
As we know that the number of constant in the general solution of a differential equation of order n is equal to its order.
Thus, the number of constant in the general equation of fourth order differential equation is four.
Question 12.
The number of arbitrary constants in the particular solution of a differential equation of third order are:
A. 3
B. 2
C. 1
D. 0
Answer:
In a particular solution of a differential equation, there is no arbitrary constant.
Exercise 9.3
Question 1.In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
![](data:image/png;base64,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)
Answer:The given equation is ![](data:image/png;base64,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)
Now, differentiating both sides w.r.t x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, again differentiating both sides, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ y’’ = 0
Therefore, the required differential equation is y’’ = 0.
Question 2.In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y2 = a(b2 – x2)
Answer:The given equation is y2 = a(b2 – x2)
Now, differentiating both sides w.r.t x, we get,
![](data:image/png;base64,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)
⇒ 2yy’ = -2ax
⇒ yy’ = -ax -------(1)
Now, again differentiating both sides, we get,
y’.y’ +yy’’ = -a
⇒ (y’)2 + yy” = -a --------(2)
Now, dividing equation (2) by (1), we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIMAAAAwCAMAAAAFHTmPAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AAAyAABmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpCQOpC2OpDbUTEAZgAAZgA6ZjoAZjo6ZjpmZjqQZmYAZmY6ZrbbZrb/kDoAkDo6kGYAkLbbkNu2kNv/tmYAtmY6tpBmttv/tv/btv//25A627Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///buXjVhgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACnklEQVRYR+1Y21bbMBCUubklJZSmLVBwoBhKnbaxsf7/39DN65UsxytZD+QUPQQDmtVod7TOiLH3QcnA329ZtqRMnDuHPy1+9zF4cbytzn6pP7Rf79nm4IG6gIQy9RE6ePEZg9rVCWPN6W0XpvlA5qCg6iNw8OLCQtRq3/3Klf3vXdEVVOPDRn2ECiGglU6l+cE2WA68gPT4FlGYDhjAghdO6kq9jElEZSlygoOCGnwABdbkEsiLLDveNnl28PNKi0OXqBYU6n7viAMgDr9Y0NbgCRz45qM+daZ8Ws24lqXSVyaGl4PRv0AMoYT11S5PtrzEEqqkLCpUl9JzxKxaAGIApXHQJZdlUAHML3yNJO1yKGVK1DDCl3CFgIfJtQcxNIcu/VKbNd76ZB4AMYBOctET+J/vuao0SEAkxJK01IM77HMBCBdK4yD1oPOgz4UY7er8EnWKduXpTTYHQDhQGgW9e7V83x/KDK/qbdFOfwCEDSVyaPIL9vJDpQD6pN0wve3O4TACJXJgj6I7/Ftl0IwkGSwA0psKEBaUSgHPk+9Nccasg4nemyMhAeFCYygISTwtnovlHUpD9/1hRzzeIeAhbvEeJbr/Wdg3D0CEQ+eSfcfvZwaGbyj/PuBlmPRhJGf+NfYzwQGsqbUICPmGpkb5qMT8Y3xUYgpRPio1hwgflZpCjI+axWHMTMUH7byVOLfiigH7p9GY4KFizZQTufdW0mBN2NcOm8RMOUTMd2mRB9q1QoSZmiyUthNN/ol4yzTbTLnF6LyVcJnky5KZZsqjB2Or2str5/ZlNIEzzZQTF3mr9QO9484zUwM5Gm/FS6HH2nJ2O3Q000w5kY23uinEDYK8YaGdjHRmavLAeCckNlNRJNKbqXAa/42ZegXMRVJZJEocmQAAAABJRU5ErkJggg==)
⇒ xyy” + x(y’)2 – yy” = 0
Therefore, the required differential equation is xyy” + x(y’)2 – yy” = 0.
Question 3.In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = a e3x + b e–2x
Answer:It is given that y = ae3x + be-2x --------(1)
Now, differentiating both side we get,
y’ = 3ae3x - 2be-2x --------(2)
Now, again differentiating both sides, we get,
y’’ = 9ae3x + 4be-2x -------(3)
Now, let us multiply equation (1) with 2 and then adding it to equation (2), we get,
(2ae3x + 2be-2x) + (3ae3x - 2be-2x) = 2y – y’
⇒ 5ae3x = 2y + y’
![](data:image/png;base64,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)
Now, let us multiply equation (1) with 3 and subtracting equation (2), we get
(3ae3x + 3be-2x) - (3ae3x - 2be-2x) = 3y – y’
⇒ 5be-2x = 3y - y’
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJMAAAAsCAMAAABWp3qHAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkLb/kNu2kNv/tmYAtpA6ttv/tv/btv//25A625Bm27Zm27aQ29u22////7Zm/7aQ/9uQ/9u2//+2///b+JBolgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACXUlEQVRYR+1YbVPCMAxuh0J9F99hikp96VDmRv//fzPpNmDHmrVu4/COfOC4S7o8TZ4kbRnby3+MgA77813DvRgOuoakZ8ecH07d3cSBh7H7Z9ctZfDM0tDDj+o+dRIzofjYeUfS3dT5m1WGsvepQ87780TwypCttK93W2F4Ojl9ZyyrJxtbaG2jgFQsXgw5P8W9q94n/NiKita2DYqxL4FwEjFm+jEvKskLyXNZ0rYPYfOLMb/E5A1YbC0qWtsByEQAJkweUVS0tk1QOkQ0sekFi+H5LeTQIrS2XUzAlzQfYRJTaBVa6wTq+5rzs3rLdCLALus5MVLdKrS23hMk4uqZzTxGhsFETldaW2BSnNwYS46mbCaCaSLqZxR0gWUj2NwzrS3Zm0ZmF2X425+rPDeUrQ7PnuxhorU1mLAro2AdzTI6qYMHh2TDRDPdvFpoLYvW5jIZJ5UzPDquz5wDaMoEm/ySUL2PCQ9MGJLr/E+hiwFSDKY/Lzo8hBnbqahVNakAJro5/CRiBGNrLYRZEscsEgMt4RTSLqSCIsupuDriGHjmgCzBK44kWjYma2tIdViUW8YnwLMYYokhME9ZbdTn36aTSkzmk3Rr8IRLmVO5K8ep1mlnuStx3PDp0oFKtXibGJR6AR/M9RsSPeYjpiOX5tjEt21tqWdeTIoLYwQ3jntvitMAo85vlz4BgjsViOexwsfBH2zXCfKH5Z0s2WNyC+tuxulEeL0AuW21kZV+HPm9ADXy5rE4u4nulmzhldBnwxIjtGNxkuZ2vLWjj1O40hsYLfhStpcmEfgFaxwzvrKGBZ4AAAAASUVORK5CYII=)
y” = 9.
+4![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ y” = 6y + y’
⇒ y” – y’ - 6y = 0
Therefore, the required differential equation is y” – y’ - 6y = 0.
Question 4.In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = e2x (a + bx)
Answer:It is given y = e2x(a + bx) -------(1)
Now, differentiating both side w.r.t. x, we get,
y’ = 2e2x(a + bx) + e2x.b ------(2)
Now, let us multiply equation (1) with 2 and then subtracting it to equation (2), we get,
y’ – 2y = e2x(2a +2bx + b) – e2x(2a + 2bx)
⇒ y’ – 2y = be2x ---------(3)
Now, again differentiating both sides w.r.t. x, we get,
y” – 2y’ = 2be2x ------(4)
Dividing equation (4) by equation (3), we get,
![](data:image/png;base64,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)
⇒ y” – 2y’ = 2y’ – 4y
⇒ y” – 4y’ – 4y = 0
Therefore, the required differential equation is y” – 4y’ - 4y = 0.
Question 5.In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = ex (a cos x + b sin x)
Answer:It is given that y = ex(acosx + bsinx) ------(1)
Now, differentiating both w.r.t. x, we get,
y’ = ex(acosx + bsinx) + ex(-asinx + bcosx)
⇒ y’ = ex[(a + b)cosx – (a – b)sinx)] ------(2)
Again, differentiating both sides w.r.t. x, we get,
y” = ex[(a + b)cosx – (a – b)sinx)] + ex[-(a + b)sinx – (a – b)cosx)]
⇒ y” = ex[2bcosx – 2asinx]
⇒ y” = 2ex(bcosx – asinx) ----(3)
Adding equation (1) and (3), we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ 2y + y” = 2y’
Therefore, the required differential equation is 2y + y” = 2y’= 0.
Question 6.Form the differential equation of the family of circles touching the y-axis at origin.
Answer:![](data:image/png;base64,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)
The center of the circle touching the y- axis at orgin lies on the x – axis.
Let (a,0) be the centre of the circle.
Thus, it touches the y – axis at orgin, its radius is a.
Now, the equation of the circle with centre (a,0) and radius (a) is
(x –a)2 – y2 = a2
⇒ x2 + y2 = 2ax
Now, differentiating both sides w.r.t. x , we get,
2x + 2yy’ = 2a
⇒ x + yy’ = a
Now, on substituting the value of a in the equation, we get,
x2 + y2 = 2(x + yy’)x
⇒ 2xyy’ + x2 = y2
Therefore, the required differential equation is 2xyy’ + x2 = y2 .
Question 7.Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Answer:We know that the equation of the parabola having the vertex at origin and the axis along the positive y- axis is
x2 = 4ay ------(1)
![](data:image/png;base64,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)
Now, differentiating equation (1) w.r.t. x, we get,
2x = 4ay’ -----(2)
On dividing equation (2) by equation (1), we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ xy’ = 2y
⇒ xy’ – 2y = 0
Therefore, the required differential equation is xy’ – 2y = 0.
Question 8.Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.
Answer:We know that the equation of the family of ellipses having foci on y – axis and the centre at origin is
-------(1)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAM0AAADPCAMAAAEWrZXGAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAH4UExURf///+Tk5MTExMLCwmJiYjQ0NFxcXLOzs5eXl6KiojIyMtDQ0NLS0o6Ojm9vb3Z2dpCQkPf39wgICAAAAIuLi729vRISEuLi4klJSRYWFmxsbEhISMjIyB8fH9TU1ExMTKurq4+Pj6+vr0tLSwMDA3l5eQ8PDwoKCoGBgSYmJry8vNnZ2dra2iEhISUlJevr65qampiYmJ2dnXJycq6urtXV1fz8/IqKil5eXnBwcICAgI2NjZmZmZWVlZaWloWFhWBgYE1NTZSUlKioqDAwMD09PfX19e3t7YmJiVNTU6mpqWVlZdfX1y0tLZycnGZmZjY2NuDg4Lu7u3d3dz4+PsfHx8XFxW5ubri4uDs7O3x8fFFRUX5+flJSUkJCQnt7e2pqaoaGhlVVVYODg7Kysjg4OPDw8Obm5pubm39/f3V1dXh4eKGhoUVFRaampjo6OszMzHp6es3NzVRUVE9PT5KSknR0dGNjY56enhsbG21tbSsrK5+fn2tra05OTsvLy6CgoKOjo8nJyRkZGXFxcaysrOnp6ZGRkVlZWcHBwba2toSEhJOTk1ZWVoiIiGlpaWRkZLS0tFtbW9zc3LCwsGhoaPr6+kNDQ7GxsVdXVyoqKmdnZ7+/v4KCgkZGRqWlpbq6ur6+vkFBQaenp7W1tYeHh62trQAAANABgCkAAACodFJOU///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////AJlf7PQAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAjdSURBVHhe7Z1NsqssEIadZAVUOXYD2YFzh1aFHTi3Kitx4oYvCIl//LxIg+ZcnzrfZ06u2jQ0TTegpwCp9dFEqY9BdIP+8CdZVBcbm1HQjO9m+sCn3zZ1xvXxR3nqIw1tUelPRDz00cQh4y2GcI17fQwCsQumj9TImmHCZBtpwOJ/TVNMH8fRIPGQdoL9dS99pKbVxw30buhPO7aD6B74VodMHLPJo5bsQ1cBS+Hq/fU66uMK32VLS3adu1IJ7b+6RODpqVrFwK5EnT462QRIFm/j5IDnMzbrzS/DDoUkR4KAC1/D2RA+jEo5Af1uEjCVDb5IxaVKnzCPpOsAC2z1WQfq7XsNcu3H38TIcSaGiq/7/F7T6KOX5lDZ4vqPz9vPTThf4xvT5zNd+jQrdz4bpeWat2y/YVXcuTFcctbM11jqrZpK0rKXzF/Vz6AO4ygSBPnxLf77fCUpDT3qU0dcZYtQGKS1EIepbFDv+fiNWl09xS1+I/0w1xteg/OZsGEH3H12NAHXfPleAwUxihA5n9r+XBPitSP0AS9VWXqgnHlcCEBG9/Ia3NIknbgm7ApJaNkkua45MhMUHNsLxxdcNtGqR/T5L8iWlkunn6MVhEUNDIu1YhlZphyMdo7VTi456TP6ehrzJnsbk2lV7aJo12zzURbD6qL/BCQ7GMsus+qnpJ12Hfqtbx0SP3jYtAOpCgu2dpxIzi5e3skJDyQg9vpQ9KV9+G+ot3gPvjanRkbBhiwn3ugM2YzJDlIMSSY5sS1kciwmOciMZScM06a3yVWa5EA4lDY1sFGOtQ8N33+Sq2D64w5YjsPi5n+q+tJSGab6NA475opnbJsNGuxXEllvDi3XrIsjrE8ImeVYCneAtRxRaOG8v/6gohu5TQZiml2JHsANNWxqn+gaNBTUIIcgEt57FIMcgrhnX3F7ObD5utiFuab2oWDbxjs5JOoINoK2cuh667rqNnKQoQdl5RZWcqjq7MNC0lIOtRjBV9Ish65lVtTqvlqOZRKZioqXPEFtGUisyJeNXSfjr8gZq2IQYQ2zxUZ0qKAmlx3gawc3N4cRZpZlujdFqmuAWzM2WkRo2xOEnR6EbyvS+7UJlsevjZlmx29u0jMwXr/qLu3SxXPuMm0y78bXg0AijXZTcq8Uw8J6VkdBn+uYq4hajm0KitYQrFOlpGIcN6O0N1c6SDdkO0tMNxnhzgKoxmyKFRwA39hPM9/qtSWaeNffNSh8AdABKcQghkSQkCK3MHnvMKAxJd4TbOLyOlGGs3Zn3FKH1HNglvvFOgqwjmJNGmzcWIvO5DbRto00QHTQog+ljPjT3rqWP5EAkQd7W0cuOJu1VtriDvZ2hlvWfovZrdrPgSvTcaIeXF/FaO1dBGLUHnQ38WKGovnK4SFr70YsYtjyCcD6ycpi2jKvts9/fpqRr74QjON7nE7VXxXqOA7iS3ncwMrnqoNbih1baWBtbH2NXjPaQxewC4SQZfkWBkoqRlTN8n7d3MdJp/I2gdKiW0eGUJumXf+6aLnIgQAtJexjzWRarwEHabh/xRGdSWE5UnSsjuX9sXEa1r3pknYnBEku0vFAe3QBNE6mOsPMxIPfWOGh3IW3h0e6zQ++2xAYgMRzGyJlfDcic5tOgyUxM0WeRTtX80T75iXW4YA4BbaoQ6qLxHhDMlueMQw8hEY2U20rLpX9raKpA8/kwzw/XT75i8HqruUknv/m5ubnkAH5UFQs9ZCt/FhyMWrIeZIPmRvU/W+/eXNz81/STVM5qR1tNqpn8VZ5XuqIOw9MTRsOLNMkVVp61TRPlmnROQc9eyZ5/eUp8FK0zd+wNUWuKaw83NpcF5p9f1fhtrRr0tR1zXhdV5ELymdTPR96blS3TTN0z59MxN9du1jaXXqBpu1/ayRtyk0gs+03w+IxjovD9yGmwQv0PxG5cdNyi9GntZfXZ/1u1y8WD33xP6hiKbX1++IRvQskGdxqOlZtHNeczMO+9ceuTba90GFUrsVWlzauWjiLwWkxTm2K7mrxQe3ekuXWJttT3iDW3asajzZFpjeYYHhaxq9NwqdUg3l7d1t4tfnMvF0Af1n9ZyCnZKHzVytQ1Pc1pt9roAcjFT9cousgW6YhM7rCxA60twfSpjo/ZGtc/uz11kE/1sXP7zmtI8cXpdN/FBHT5nW6Oq6mGfpKj6uYNml2CQYAPlgHaoO4x5SAfgjU5my3Blo6qk13amKNjnioNuea2nI+0wWqzegLxpOCPjqFaoOf6KIXFiMaOThUQgcIuJA0HWc4lF/AsQiszeYNQAAVE01QrZuBN0faGO608M2PPBjeb/azT290KMP9CSwb1uZIWvDqaHwHrA08A3jgTRnC2l/mUbftgwwOfmQBb5sQA2HSeRnhT1ayJ2OMt23Xgjx6/cFBx3nHmf7FQ8cfD951HCiBPEOUtiwZs6w01K1QR3+GgC0NfqQ3qN+U5dOVkIQCe2hYmyM+jQp3ICKb2ZHfGD1De+a0misWKCf3pJYDDdoMg2n2+dR8zWVB3aSHeprKcJ5wqoaWPXUxxzlhIxdBWnXCvm1ECGNomjfaEZOwezLPAuoF5LuuTwS0c1SbkzPpFtuTAY5i5yZr8IgDanP6M5FYOUFtwNPSgaU4WDEvsEoAdVxMm5N9gATqOZA2l1iZRjYvINo0F2gaAVAKRJvTFwgU7m0cE4A2l9kyoIMxB35tkLe5ZcI7GefV5iKdRuErrVcbwBQz4gkro7XNjDvN8pQWDbHzoR+4M+PUZrxay0hah4t1lXe4jjdbov76sRGHNv3ZWYCV0mZtVm0sM8jXwGY1Nm2sE8kXQc07bTFrwy8Smrkw/bUfkzbtNXv/jmHXGfba/NLDuU27NqKNNkSLYTnhi+pfaLP8+seoOOuHYSzY+H7VcqnpsoNLCGaf9qtc8jmOw/yttrm1uS63Ntfl1uY6DNOjaG01rWPWXO5M+eXn8eteqFQU7ykZlRt4fi/OXLHY7lax5O8qTE23SJI5+/Voc9VPftzObm7+LkXxD62rl+Er/meaAAAAAElFTkSuQmCC)
Now, differentiating equation (1) w.r.t. x, we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGoAAAAsCAMAAAC6/t3UAAAAAXNSR0IArs4c6QAAAKJQTFRFAAAAAAAAAAA6AAAyAABmADpmADqQAGa2OgAAOgA6OjoAOjqQOmY6OmaQOma2OpCQOpC2OpDbUTEAZgAAZgA6ZjoAZjo6ZjpmZjqQZmYAZmY6ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNu2kNv/tmYAtmY6tpA6ttv/tv/btv//25A627Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///bcQFijwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACaUlEQVRYR+1XC3PTMAyOw2jDYxslwHgGNjDbcAdxGv//v4Yk26JtWlvxjd1xh++2addP/uxPD6tV9X8dU8B1i/6B1BnbpYTJ3b1W6vG3LDSFs3XeHwi0+lht2pMfOa4Uzsj003h3oz5lqRI4nfX+s7sFKq3UojcqSQo41yFuaNSjl8Gov76dkRQaBdSguOuSB0ScTzcIDxs5QbY/HxokQFezSvl5nMFzGdCTDTmX6zzB0Dw/S3kFHBK6z5BzbIipXBerwiSTNuLwr0UR2RBT6WWI6njxLpX0jAPhfM6xIeSiorAgIYiSKnvGVWN7fkF1yIaMangCqju9op/KqmN5wTgq+4BiQ8QF5YRrBfVSw63AOuwWcfipjTqzIaIqAdmYR2yU7JL1gYhSpmNovfHXluvOvtCl2MhQuUulnt3kQIe6FMT0lIqDjfQu2IM2Xe7lyDREmXD0lmE7T657oaJOnn2m74PKt/+xzTyeBVSh+qBQQzv1VPTbF/DOIlH3fQ7gDnhN47FFlQpWwa0m20UBMyNVAdVEwCqkRfLJRYVnjCfH5KEJ6WGSHZPPq2gUF/JPmD13Hvz0rSbwI/fawLa+v9Asgmt8dVWtZWNqAXyHCv+h90++ZsL5VsSbSZS9U8yFn9xeqvoDbbJOzmaex62fxpBK4NuHM/XpTZjZjYSpW/YO52CQQADfkYFmP+q8FlytqJJoypXDI5+PlV70OMKkvxzwEZFqBnxCJUw8d/emEZ5of0e+lYwKhueehvWCFWIlzXJ6v0upFKTUtbhLDM2q2rwvvNULGJ4EX7ujYN+hqn61ubGkQN5/0+U3z+BE1+MVWxsAAAAASUVORK5CYII=)
----(2)
Now, again differentiating w.r.t. x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMwAAAAqCAMAAADI4URfAAAAAXNSR0IArs4c6QAAAKJQTFRFAAAAAAAAAAA6AAAyAABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmaQOma2OpCQOpC2OpDbUTEAZgAAZgA6ZjoAZjo6ZjpmZjqQZmY6ZpDbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tpA6tpBmttv/tv/btv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///b6jjBjAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADPUlEQVRoQ+1Yi3LTMBCUQiEEaEqLaXmZhhpoESnYjvX/v4YelizJsrXpRNMwU02nk7R7pz3pbnUSIU/jv12B+6sbhDsIQ1xlw+wu6cmvtHcQlnaUE9GubxkQDAjLyRTzjQQjPIEwbM5sKJAlCMtGE3MMsgRh2JzZUCBLEJaNJuYYZAnCsDmzoUCWICwbTcwxyBKEYXNmQ7EF1AGAsGw0Acd8s6KUPv+cgIIw8ueS0jNg2qOD8J+nbh/UFcu6e/+NbLGNHsJh69tHj42X57VLol19UV/bV1DWSmi1rMUPad9oy8cbvLzwJ2/6Fo8Ff5+hWL0k4icdP99Qevrw7eNlarEMd8uVSVqEbN2SSbgxwRAm9md68HJZ78pIO/w7RVL7TAbDS83djt6CeeWfcCM2Ue/jfGo2sgobOmZuMjuRn8lgtB9eUrqs2xVd3HQf5eI2IpZmmNVxY6HP3lkbZyXmcrOSm9IVwepJY+y4SAajVktGIxOk/yJnFDrvLKHrxkIDGxWRKp2JofFSLHv/cg4zEOlEg9G3nL5cxmQ8NxYasZGipkc14qmD0b/DwU0puQH2n6PuRji1wX09ynTjX2NiPGJloRGbIZgYYSCY6Y0FBMBklhSCZlKLvJ2x0IhNOhhVMzqNs6WZ3KJqUiH9bLXQsc1QM+M0I70ARDTiQAJgVbEr3n6YfN3xg7HQkU1XzKkZk5KSU5qHc6ai00QCHbHQ0Gb+nJFCFq1/59Bkw4PZqNmNq5kLsx3AqBVwajFwM2kz3wEQ8WhH1xEtc6veXrfAZteD2d6smTkiAo2x0MBmj950UrW8uyPocIDJrlmIclyXIwJroGObg3TNXjBgs+vAxH3mrjy7BjeGG6j9YAI+zH2Gndxt6EJfMr1mN1hXvn1t7o8BTDRcqVw2vix0D5u5kzD8H1uIO57SPeI3uz5QCFfNK9UEzcH2mTkDVh056mQNmt3IXOpcScMysARd6poRnUTY7E4EA8DAmTPAbDAJ3/z+SjzhYLe6DDQxl2AwsmYIeKnDJs6B6msm9QChGuTjD4YKmfqRvKu1qwuy+9Q/JOVY1IP4ZOfiCedF+pXruzhl/hbHXjQHWZInJw9bgX8sf18h1jDkawAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Let us substitute the value in eq. (2), we get,
![](data:image/png;base64,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)
⇒ -x(y’)2 – xyy” + yy’ = 0
⇒ xyy” + x(y’)2 – yy’ = 0
Therefore, the required differential equation is xyy” + x(y’)2 – yy’ = 0.
Question 9.Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.
Answer:We know that the equation of the family of hyperbolas having foci on x – axis and the centre at origin is
-------(1)
![](data:image/png;base64,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)
Now, differentiating equation (1) w.r.t. x, we get,
![](data:image/png;base64,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)
--------(2)
Now, again differentiating w.r.t. x, we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJQAAAArCAMAAACpflFGAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AAAyAABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpCQOpC2OpDbUTEAZgAAZgA6ZjoAZjo6ZjpmZjqQZmY6ZpDbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tpA6ttv/tv/btv//25A627Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///b7KvNqAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACq0lEQVRYR+1YDVPbMAy1y0bbjTIgwL4CXQMFly1Jk///42Y7sWMrTiw75cpx5HpcVZ6kF1mWZBPyLp4qmefmi0D5KC9ZLn9bfqE8Riqb5/yDpv1yu0Fii0/PFhLKo6QWJFsg/ZD9DQWuhjUZsGrKdWpHEVrhjNCkytWWYUlBv5bsI8UuCf+gn3FSdUrpPC+XdLapfvCcgLJyY5LSmJMrrYym0wA9kapTkZ/FTCUelFtvIGytzgDYz9C3fPL/RvpAufFgL5/GuMHTSYmNX991OxTKhGRUPSqeGtMH+wkhlo/HYEEKo8RA2RUpjXGD/cR8yyeSLjP3O5Qdy9fpOMEHIFUl376bVRPKLlIa4wQjSOmNNYTNqF1hoNxPdJFnrY4L7CFVr5c8ST//GoVh2gssnlonpBf5I9ghCtBeoOyypTEYcAgZuSx3G7Me9GSXPa0DlUOdD+Hr9PzeDBSUnaSUDgbcVpQ1pWdbLGnex1bmGARlN6lWBwNu9+4836fYyQDLfSJOttaCjk8/E30Eq2ciSFWCngKDHUQoNMPEcUf+XgNvSMm/urerL/IdoUYPNeWHgSgapCLi/EoqavmOmVP9+atN9IAh/pXCY5plohq8tZIgNl6zht3z94bS85iAsO7QGG1D+uXHT7tzkOr6D9l5pyonaT22TrAxHI3yC/bwbtmwZulIG8OkQo6vhhWLVKQNm1S9+6pyaReVUuIk+7Sms2ZwjbVhkeLHn7zOZC6xSE6EzVZbIjd0vI3+8sk7poJzKqLmBibeSbb3eBtuUlUi+lkcKTmY8euwCTZATr3c8rNMFBllqEn0kDs6TzkUOUVCrggd9g5OSs6gU0k1OXW4TlouL8n+J7hcDWw2jPId/BjXDdyuHniV+pdMSip2wc9Gp1HNIPD1P+B2BP4DV4VJgT2o/JQAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Let us substitute the value in eq. (2), we get,
![](data:image/png;base64,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)
⇒ x(y’)2 + xyy” - yy’ = 0
⇒ xyy” + x(y’)2 – yy’ = 0
Therefore, the required differential equation is xyy” + x(y’)2 – yy’ = 0.
Question 10.Form the differential equation of the family of circles having centre on y-axis and radius 3 units.
Answer:![](data:image/png;base64,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)
Let the centre of the circle on y – axis be (0,b).
We know that the differential equation of the family of circles with centre at (0, b) and radius 3 is: x2 + (y- b)2 = 32
⇒ x2 + (y- b)2 = 9 ----(1)
Now, differentiating both sides w.r.t. x, we get,
2x + 2(y – b).y’ = 0
⇒ (y – b). y’ = x
⇒ y – b = ![](data:image/png;base64,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)
Thus, substituting the value of ( y – b) in equation (1), we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ x2((y’)2 + 1) = 9(y’)2
⇒ (x2 – 9)(y’)2 + x2 = 0
Therefore, the required differential equation is (x2 – 9)(y’)2 + x2 = 0
Question 11.Which of the following differential equations has y = c1ex + c2e–x as the general solution?
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:It is given that y = c1ex + c2e-x
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
Again, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the required differential equation is
.
Question 12.Which of the following differential equations has y = x as one of its particular solution?
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:It is given that that y = x
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
Again, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
Now, substitute the value of
in the given options, we will see that the differential equation in option (C) is correct.
![](data:image/png;base64,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)
= -x2 + x2
= 0
In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
Answer:
The given equation is
Now, differentiating both sides w.r.t x, we get,
Now, again differentiating both sides, we get,
⇒ y’’ = 0
Therefore, the required differential equation is y’’ = 0.
Question 2.
In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y2 = a(b2 – x2)
Answer:
The given equation is y2 = a(b2 – x2)
Now, differentiating both sides w.r.t x, we get,
⇒ 2yy’ = -2ax
⇒ yy’ = -ax -------(1)
Now, again differentiating both sides, we get,
y’.y’ +yy’’ = -a
⇒ (y’)2 + yy” = -a --------(2)
Now, dividing equation (2) by (1), we get,
⇒ xyy” + x(y’)2 – yy” = 0
Therefore, the required differential equation is xyy” + x(y’)2 – yy” = 0.
Question 3.
In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = a e3x + b e–2x
Answer:
It is given that y = ae3x + be-2x --------(1)
Now, differentiating both side we get,
y’ = 3ae3x - 2be-2x --------(2)
Now, again differentiating both sides, we get,
y’’ = 9ae3x + 4be-2x -------(3)
Now, let us multiply equation (1) with 2 and then adding it to equation (2), we get,
(2ae3x + 2be-2x) + (3ae3x - 2be-2x) = 2y – y’
⇒ 5ae3x = 2y + y’
Now, let us multiply equation (1) with 3 and subtracting equation (2), we get
(3ae3x + 3be-2x) - (3ae3x - 2be-2x) = 3y – y’
⇒ 5be-2x = 3y - y’
y” = 9. +4
⇒ y” = 6y + y’
⇒ y” – y’ - 6y = 0
Therefore, the required differential equation is y” – y’ - 6y = 0.
Question 4.
In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = e2x (a + bx)
Answer:
It is given y = e2x(a + bx) -------(1)
Now, differentiating both side w.r.t. x, we get,
y’ = 2e2x(a + bx) + e2x.b ------(2)
Now, let us multiply equation (1) with 2 and then subtracting it to equation (2), we get,
y’ – 2y = e2x(2a +2bx + b) – e2x(2a + 2bx)
⇒ y’ – 2y = be2x ---------(3)
Now, again differentiating both sides w.r.t. x, we get,
y” – 2y’ = 2be2x ------(4)
Dividing equation (4) by equation (3), we get,
⇒ y” – 2y’ = 2y’ – 4y
⇒ y” – 4y’ – 4y = 0
Therefore, the required differential equation is y” – 4y’ - 4y = 0.
Question 5.
In each of the question, form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
y = ex (a cos x + b sin x)
Answer:
It is given that y = ex(acosx + bsinx) ------(1)
Now, differentiating both w.r.t. x, we get,
y’ = ex(acosx + bsinx) + ex(-asinx + bcosx)
⇒ y’ = ex[(a + b)cosx – (a – b)sinx)] ------(2)
Again, differentiating both sides w.r.t. x, we get,
y” = ex[(a + b)cosx – (a – b)sinx)] + ex[-(a + b)sinx – (a – b)cosx)]
⇒ y” = ex[2bcosx – 2asinx]
⇒ y” = 2ex(bcosx – asinx) ----(3)
Adding equation (1) and (3), we get,
⇒ 2y + y” = 2y’
Therefore, the required differential equation is 2y + y” = 2y’= 0.
Question 6.
Form the differential equation of the family of circles touching the y-axis at origin.
Answer:
The center of the circle touching the y- axis at orgin lies on the x – axis.
Let (a,0) be the centre of the circle.
Thus, it touches the y – axis at orgin, its radius is a.
Now, the equation of the circle with centre (a,0) and radius (a) is
(x –a)2 – y2 = a2
⇒ x2 + y2 = 2ax
Now, differentiating both sides w.r.t. x , we get,
2x + 2yy’ = 2a
⇒ x + yy’ = a
Now, on substituting the value of a in the equation, we get,
x2 + y2 = 2(x + yy’)x
⇒ 2xyy’ + x2 = y2
Therefore, the required differential equation is 2xyy’ + x2 = y2 .
Question 7.
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Answer:
We know that the equation of the parabola having the vertex at origin and the axis along the positive y- axis is
x2 = 4ay ------(1)
Now, differentiating equation (1) w.r.t. x, we get,
2x = 4ay’ -----(2)
On dividing equation (2) by equation (1), we get,
⇒ xy’ = 2y
⇒ xy’ – 2y = 0
Therefore, the required differential equation is xy’ – 2y = 0.
Question 8.
Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.
Answer:
We know that the equation of the family of ellipses having foci on y – axis and the centre at origin is
-------(1)
Now, differentiating equation (1) w.r.t. x, we get,
----(2)
Now, again differentiating w.r.t. x, we get,
Let us substitute the value in eq. (2), we get,
⇒ -x(y’)2 – xyy” + yy’ = 0
⇒ xyy” + x(y’)2 – yy’ = 0
Therefore, the required differential equation is xyy” + x(y’)2 – yy’ = 0.
Question 9.
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.
Answer:
We know that the equation of the family of hyperbolas having foci on x – axis and the centre at origin is
-------(1)
Now, differentiating equation (1) w.r.t. x, we get,
--------(2)
Now, again differentiating w.r.t. x, we get,
Let us substitute the value in eq. (2), we get,
⇒ x(y’)2 + xyy” - yy’ = 0
⇒ xyy” + x(y’)2 – yy’ = 0
Therefore, the required differential equation is xyy” + x(y’)2 – yy’ = 0.
Question 10.
Form the differential equation of the family of circles having centre on y-axis and radius 3 units.
Answer:
Let the centre of the circle on y – axis be (0,b).
We know that the differential equation of the family of circles with centre at (0, b) and radius 3 is: x2 + (y- b)2 = 32
⇒ x2 + (y- b)2 = 9 ----(1)
Now, differentiating both sides w.r.t. x, we get,
2x + 2(y – b).y’ = 0
⇒ (y – b). y’ = x
⇒ y – b =
Thus, substituting the value of ( y – b) in equation (1), we get,
⇒ x2((y’)2 + 1) = 9(y’)2
⇒ (x2 – 9)(y’)2 + x2 = 0
Therefore, the required differential equation is (x2 – 9)(y’)2 + x2 = 0
Question 11.
Which of the following differential equations has y = c1ex + c2e–x as the general solution?
A.
B.
C.
D.
Answer:
It is given that y = c1ex + c2e-x
Now, differentiating both sides w.r.t. x, we get,
Again, differentiating both sides w.r.t. x, we get,
Therefore, the required differential equation is .
Question 12.
Which of the following differential equations has y = x as one of its particular solution?
A.
B.
C.
D.
Answer:
It is given that that y = x
Now, differentiating both sides w.r.t. x, we get,
Again, differentiating both sides w.r.t. x, we get,
Now, substitute the value of in the given options, we will see that the differential equation in option (C) is correct.
= -x2 + x2
= 0
Exercise 9.4
Question 1.For each of the differential equations in question, find the general solution:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
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![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPAAAAArCAMAAABxcvVMAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OgBmOjoAOjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkLbbkNv/tmYAtmY6tmZmttvbttv/tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///bD1pXxAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAD5ElEQVRoQ+1Z21LcMAy1wyVtKV1YKG0hXEJbArQkm/z/x1V2somV2F5Jpn0pHobJwOocHVlWJK9Sb+t/j0B3r/VpWhBeAYLjQCJdeargJ2m9AgSHP42uef/AIfN9lglRan3wUml9JeRl0s1ZyoMXIfFoxoUoszvVFVK9ikuH5XVFYjor8J0J0RWww0ybyWs2HRbc5GLmLRAfosmPV+K04tMhqlp8lEYYAUQFSS1dAjqXqkoXzIdoL77siUslnw6FNiXWAxAboru+a9eH0h1m02EiUzETFxeiM6/9WtzscOnmRVqeWwNSV9Agfp9rbUpVV+gMdlja3lHpAruYaG5QiRDt2a16Sk8nKl0oa9v1bHtqdge0gAifkOZd+PxQiZd0VEvrV7ueN1rsqr+ECCqOdhtEYg8d0TIguMmZPR9d8FO02yASe+iIllZwk893mGMdgAjscBXvrojES49BhHePmqOfS0+weQOF9Di/gjoK80yTQzUlvLEwxOZa62wF8whgZd9sTAE0A49q0FsHsmcgVoRBaiY4Zll7/K+10wA0+aXqbqD1Mt09eEfSC29UB8JYdjdgbbCeTeTNw2Z9BdVCw/ILHonV7kEK0Vl06/Jk2RP1a6kAmdu5y1aAytTuym2GIPYBEATRrqGpaD6+2BmuKwCBMsxNxLsHKSx4h2VlUgst19w62x8I8wsaQEJCmwi5gbnXRz9s9TdY4E7/gNa0AcOfHWJgngYpb5CX8R3OsG8EIws2e1MTbwawYPX0Qe8/DAms9x48ghdhdAWrXa1yUPDWMp7SldPSuryQ0yXKhnBKuxBWy3N+OOzwuNXxVHGJdw5SiC5u6StCrrk9cv0ZBodPLogDHPLAHF+TyT0WrPEhotkh3j1IIbqopfe1hMyr7FZtPveVtCRPM1iwqcnnILbWUD4f4b1UmYfv8XIwEhMGKZxQHEsbcjRNw31vtvqVW8U1bQSaQ2yuc60/mWvBR3iNfx0e9m/jOb0lviQMUnj+51haH8rQhUdNHtCDEEGR3TP0IvvxTQ8a8+kQlHeahoNEfSeZmNH6E4e2hCzfzMe0eA5M/+XTEQQXqxvyBksEG3Dp1VSaYP/wDr20PYakRZz/F1icmc4xltINEH3TnLSkECW5KiL3pHQDSMLl4dYNIQRxGFzshZBu8jb5ewdK77hMIfH3JTK60QFpnB0FIghK/+U/aCK6CYreXgTPuQiiPJSWDhHd5HyVXLOUBMLa1KLTJKGb9CZ8SbsFkUDY21rbNrOXhG4sWGd31JE35FcrgxhGTbZgId1UoeGmjh1iZACzdioEx4F/TMdx7e2zbxH4axH4A4Isa8prIML/AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
Question 2.For each of the differential equations in question, find the general solution:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Question 3.For each of the differential equations in question, find the general solution:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
dy = (1 - y) dx
Separating variables
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHMAAAAuCAMAAAAsjdb9AAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOjpmOmY6OmaQOma2OpCQOpDbZgAAZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A627aQ29uQ2////7Zm/9uQ/9u2//+2///biJ0OagAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABkUlEQVRYR+1WXVPCMBC8FCFAtYBCRK0YJaH5/7/QS6klrdMwk3rnC/fQ4QHY7n3sLsCtOh0wQuzYW2L+AdNKfp7cmHYlxELunBJierRSZG/ko7VyC26P83RqegQwDJBQ1kh+h/TkAx8zcpZQFQ8IUs/TP9wzfWdDTKdmYDxr6gp4+uaWHDfjyTXzRM73GxwpfensFU7rsw6Vwg+Xvty7yPIvWYMav7nMZRgOJaSEV5J2KCNMyal8n0Yz3ZRQb5dpt8ltEH46A5h2fqBZx8aU8MrQlHQ3bhDZRWtKUKIfOYUHVxWirZEehUSaCv/oYkreCXVfUXQWSOnlVeKfrgwkFGsrF3n/6x3MPxpuiAn63ADq3oaY1eaxp5w0OxSYEmpYVXQ0hepWWlNyJe6PYbGlH1PaKoyLfpJXvPBzzRB9Ost6Wglus7TLQx0tmWsYky7SR3iSRfpYb6kifYD5S8CpIn2MJ1Wkj+4tUaSPYhJF+sZ7Bi6UINK7F4luffc0qAm3SE8jl+mRPv190iN9OuaYX34D+1Mh0YonuGwAAAAASUVORK5CYII=)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJoAAAAuCAMAAADnevBGAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOgBmOjpmOmY6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A627aQ29uQ2////7Zm/9uQ/9u2//+2///bbMz32QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACv0lEQVRYR+1Ya1PkIBCEvVN8rMbHneY8jR5qMPz/H+iQxCyE4TGwtX45yrIsA01nBnp6wtj/ccgI9JzfH3I/yl79t1PTz5xfYpSVqItaEDg7Pt0lgx9k1FILAudSUyf/MFrXnJ+Je91yfvSuBN885eIt83BgCkx39O5PV+KO6Qc4a7o1j/sCZgwFpjDTLZbMEXa8BvIHBFUeUyCnuTgwBUcJhNrQmH+OZ8380n/o6YSF6AEmcEMVwqKm22PWYzlP7VEvPRITL4uayWhXIiIocOp9nOcSO+EmVPNZY0NzcYvd4dQuKHBqkfO8Qy+f3Dyyj5spoB0uyKldcODUKuu5bs0N9AYI+Wb7JkZuPTojtUcAOLUsTc1G6AuUw2hH0RtZGw9NFAFko0g5GBzRFTDZyAxNVBh0u30oChpQWwNT1SRJjZ8jdSzjxPjAVLegRImcZlDzgSPU1OmLD3kgamoyMiBEYGSkb58x/9DzsqOUjJsDvBgZZtROt3N1GRq+DF9eK6lBFOaxhnaAd0bGmCyJ1X25WVdDG2H3CvG/khEzE2xguyQrcbbFAOLUsrbMnRSixqzaGk+oLCuQXwTDCbWB7agNt78QkceuQSW1cAxtYMvIQG0ZGu/moeJxEGqQxNnIaNO79XmZqjd8gcA5wF9G5q6F1sycsBxv3q1M7utNQRuAsVsD516f3TzX8H1c81ors1yQkv7Q4e9QU+cvY2+3j7F/kxumRmvk653k1JxbIxI1UiPvAZMz4WlMLKGURh4RLyK5SafxqHlKT2nkPWAisfnbQWZCwczkN/JUS+sz9xq56A0lNPJlHaITpLX9jlIjNPKy1tcvfnOhG/8ckN3I+8CkszZcPa0+Aem/Akzkz99BmLw0+cAkXjAZyiw16nmNfAEwlbs7v7yRr9s3Y3V5I58BXjcFamhhI1+377es/gTRXDpMQxhXvQAAAABJRU5ErkJggg==)
⇒ -log(1-y) = x + logc
⇒ -log(1-y) - logc = x
⇒ log (1-y)c = -x
![](data:image/png;base64,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)
Or
![](data:image/png;base64,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)
⟹![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHQAAAAgCAMAAAD0W6z0AAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOmZmOmaQOma2OpCQOpDbZgAAZjoAZjo6ZjpmZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLb/kNv/tmYAtmY6tmaQttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///bTJJP6QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABc0lEQVRIS+1V2VbDIBCFaGtwaTVxqdQaaiU2EcP//52DzTlmgTLk9CQv5TXD3IU7E0LOZ7ADehsPvjv0Yn6Xjg9KiDyDDn0xzL0p7NXiqrBz+2JRptjc8dWhZ/+Y+ZVKCufBXlfOC7kMwvxJ6cVnt5nmr14eVWJ4UAqV8vLFW94sUIudHAba6JLf/pnLKZ0XitEI41wgqEpp1FL2/a75bEeI5ga7xGCSQKWKrcieNezPWawFaASbDf1WysXhCeB0qASCCmivuXU9KeCi1whza37/L2QlV9OFqioxGTbI/WO4QJQxoQpTWoe2nwMDBa1EK/cnsveg1HGq5P6pN3/W2jCljuesOwvXAukiy37Gjy2Hkq6Izh3boLTb3oHUGwYh6W6Uoxsph/F/dqSlHP2nu86Q84KJN7aGL99GFwq7d4GaUawIojewzXHTgO7pK4RBKdTNyKDqGrVRfdzDvk8CWiVxoT9OGxO/bPhzzyZw2E9siopfQ6QcyY+NWVMAAAAASUVORK5CYII=)
Question 4.For each of the differential equations in question, find the general solution:
sec2x tan y dx + sec2y tan x dy
Answer:![](data:image/png;base64,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)
Dividing both sides by (tanx)(tany)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOYAAAAwCAMAAAAy0/a7AAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OgBmOjoAOjpmOjqQOmY6OmaQOma2OpCQOpC8OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkLyQkLbbkNv/tmYAtmY6tmZmttvbttv/tv//25A625Bm27Zm27aQ29uQ29u22/+22////7Zm/9uQ/9u2//+2///b6CauzAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEXUlEQVRoQ+2ZDXebIBSGJf1wW9dla/qxj8Z2bdqttt0Wjfz/n7Z7EfWicoFGk+ysnJ4cC8jDCxfQ1yh6Ta8jYI7A7zMhplsYlM1yi9Ob6Glyu3GdW+Dmb6wyMyHmow2BjTsSND2xK8lGlGnlBkLlnRCMBC3uiVuaeRw+m37YyM4NhC5OIvhzpJTdgAKJiuWFjRhuGDR/+1BJXF0JMZkuoyg/E5NvmAsXYgIzlYHKDGdsIcTBMqVrEascw2x2S9iRI9jIzZUJcvNY6H1QQ5ts1zRFiwPQpZJMDpbyGv7N48voGeMQL1azeVTMBCQVmAsAyaSJUawir7GsXcKTG6wXF/sGI12prKBmNkOUSR2wxQwu8/dLpVwmh9DzegjqFrBhuiuoKmo3aJewMgkWBtGDm+5B0KXQJzXWNdTIZoh53KzLO3H0A6oqLDZVXrRSHh+TdVpWKZeJWcJPJsVGPlxEyKvyTCNQms0RjW356Z3Yf9AhKvYeemVGKX1OoDLNEl6meRp4cDG6Mh1cBEqzOWJqHnnP8aEerHrUWncXF58xfnSiMs0SXmYLC1uBiwvhudBbAoWSbFZmMze4LDFW1bLEVF+QBiBwipkur6uUa9MsccgkIeHFhcH/eKGHt+xXGRAkmyPi/qiT2lfPoIVMwO75CCdKihf39CFP4hGbkceJdHITrc6B2ClhdRJsuZ87uNDYoqHWUDPbTpRJE4Grq1iID3i8PMIB9VVf7N/QuUzg5MLjpd6b4GFmMv0Vi8tOCSeTYuHYdHKxrazpaQXF6STZfjL5MBu01JDp2XLWLBV6hyXbbLSYkf3EkzdAtVAsrPvqNKF0S3a3g8Ws+wQwgAxXE6FYmUyveybTkv0Pyyw3jVaCZ9q+7G7FPN7KbI6IzY9+/g8y6+d7Y58S/fuXa3GtWZ4Nji1fosrUda2G55EBgFdQC3hULDxbt80MymuGY60rn4k2ZK5F0zcbUF6mT/8GqjP8bPJBm3q4XS+WZg/aMbGVxUC7PSaPGZ4Rsb0Hyog8Lgg2jW2/3r44QMNu3DR2MaJfzijfNJa+3vZ0i/lwEjZ9rdosdgSoQ2bfrrWWPn0zix0e6ni9ha8rAhw+Cc4bft7Ej1BgLMBVoMXefbNg3nJt0LbzHjDcpWttT2pgwWFaSjX+eXx0CwZRsMXelclhbVBvi72jh5p0fWKb+FGOs/4BIyjIYg/EWqG+FnsPj/8W1kesXFJqvgcEEFbt97mrRqxQX4u90xvXx7OSKJ/PwXurZlP30TDfA2XyWCvU12Lv9Mbl/tXLpIpX0FrKDLHYA7F2qKfF3uGlDotEEdVPszaVzCCLPRBrhfpa7N0dz/ExHT9c3X+Hn9UX3IKUn495YRZ7INYGhWaI8+67TorT2+oTk/0W8NTn8GFOTP/MxBy+Fos5vth9CrLYzdY9sP1QXCquRdYjBPq7DVdvLayXxe47z7tZz9ti383u+/bK22L3bXA363lb7LvZ/UF69RdII5krsDtaPQAAAABJRU5ErkJggg==)
⇒ lettan x = t &tany = u
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ log t = -log u + log c
Or,
⇒ log(tanx) = -log(tany) + log c
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALYAAAAqCAMAAAApilEOAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZjoAZjo6ZjpmZmYAZmY6ZrbbZrb/kDoAkDo6kGYAkLyQkLbbkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ2/+22////7Zm/7aQ/9uQ/9u2//+2///b11Y4yQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACmklEQVRYR+1ZYVejMBBMWpU7T6v11BY9Sz2xWsTm//+7m90gAQ6S9pHU8l7zgWKTTIbN7GaoQhzbMQImAvmNlKP50CKSRzPxOR0c7eR0PbRIg+9mej1A1kfae900FZ/tdT1fi6VyJtTzwhfc3nBeI3nytLfVvmuhVI4Cbk049CwkbREMvRs4/9F/H76Bto8lfWBQ/uTnf+tppIHJgP2iHrWMJBoswpI+xy9q9VPKiRCZlHOkO+7QEhqR4htXSnaiuyY2+5vPz3+zAYvHLyA0XohXTYd7cBCsVaLHnC9EWjClb1TsZK0xWtG34r2ZUux0q+cgAyd0SOXRnFiSseEzyzwg9XA3LtrzqBjRrtofxL8Nvht9K9rVQWnNjhOwNmB0LWgzoTbapVXLowutF3vrRDdRtNzVsLtoM2Xa/VVFJEK9/Ybav6JtHGa6VeE0tJvorgfmfrtITLQRbqhoxnNKbX/pA9xL2pvbO0qFsllE0o6+FW0zqC0lSamFeCuC5ZFFTjWjrR4WRQa4RdKOvhvt/wsgCSCT16gk4K5ik2eUfs9/cPm8p2zMtFK4XyX4oEmuxpNa0V0zHeHQlWUF/U7odQ7FGI1vUbmRvCjfk/epnOekcNIauEJJI0Sb760Nxb4TvR/v+uzNDdXiD4pugBYMPeU8C/VGWkX34XjKyLJe1bJWJPyFvYruy61oduRA5GXDt3gjbtAtjodq8OkauTTytq4/IIvj0RXT73b4Im5zPJwD6UG+zNscD5u6h/4vKb5CXMHRtNsdD/mX7DB/FbQ6HqgkCXN09N0Am+PBwXF1WzVqfRfzN9/ieLBI4vQN/pjshtTpeAgmC3Tg7UZx59HZQVY/62Og9B1o9bPTjiePwws2ufnLIf4np9yLf5LuS5EF9AhlAAAAAElFTkSuQmCC)
Or
⇒ (tan x) (tan y) = c
Question 5.For each of the differential equations in question, find the general solution:
(ex + e–x)dy – (ex – e–x)dx = 0
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,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)
Let (
= t
Then,![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
So,
⇒ y = log t
Or,
+ c
Question 6.For each of the differential equations in question, find the general solution:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALAAAAA1CAYAAADhymE1AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAZvSURBVHja7VxNTxtXFL1OtiHdNuDnXZptmGfvuozGg/KxsxLb8orWikZoVAHRLFAw7JCqdJ1208IGmp9AYNVNouxbtcAPoOUnBNy584En45nxm483/uBGOooYbPB5c+a+8+47PNje3gYCYVpBg0AgARMIJGACQaaArX8lC7cA+iUaOMJUCbhviVd/rj5RGHxi6soaDRxhqgSsa2zVqr6XFvokYMJUWoitNn9CAiaQgAmEogTsLNrgNv4fJuCG8z0X7255199dv29wjUAoVMCm3njAWemjJcTPwPiHdpt/7xdwr9e5xwFOSmCv8frAlg67Ozt3dna6d5YYHFqvvfKu0UATChVwz2wsKFA6Y1p7EwXYM5r3OYOPwQpsXwc4BVDOOqY5fy1+szPPgR82jd59GmRCoQLGfm9LgT1g2nt/9VxR2VqYBzY0topVmDd7T32vXWeasUoDTChcwD1DW2QAF0GhRi3iXCtxCkprD8WPFkJj7HfN6C3SAAvcAGeD6PZNXDPkwV242xDXhWhz+A2AXTzSN5QN/ZHC2NIBeV+xG9hr8qf2OgP77MqLvZuy05kX9zgBr4sK2K3a/+F7yD6Iwx433vrRHt817aE3hsQ9i4A7/LHdWbAsgYiwEY5tgPfAlE/VKjsg+yAGo9P5dmidUdaObsLslRf34SfD87SgnDVMc+H6ulPu+06pH/YrrvCvgou/caFv+6vpmo7tIhEYP4cHlGaZdxbuoRd1p7NwCWX1uNHtfrWhN77RqpWfbQHjdVYf6u96VXgS7ANupBgq/66mGcu+xYK7YJjcm4tViGn66pddntpyrbnSFFnkBHnfBO7RP9B6cwXgHE12paq+Ndq15wCV85rW/iGqwrY4ezNu+2B2G3eXGDtAf4WfE2+erje4qvJNxuDvSd0Kx955taK9DSsMdQaHpXL9qNE174ry9sQ769xz/RC8Uv9lnPYBp5wXCvvJ/ySjtYFK5c9KBf6a1CwHfu4255udXu9e1GtaHH69xZu7YVU0jPdN4Z6Dz3T6ePoEdB+wAxLlwdOGkWR7SqySOH3yztZj/5QftVAOW0TH8U7LvQgvnQf3zL7F9cVXAPwk7ikqYhrC7W9vMPISsO3NBN/zZbhp4DkD10t+kbxuLj5j6stX3vfx63gOw9v2cbzTck/Ce5jj6GBXHtwzCxinKNsnM/5h3LkHuwoBP4t6iGQL2BduunIe6tIVTunedTvwBOxfbW3roX+x7DbyfVD+ieLg77eL8i5CwGmCXXlwz9UD52FDRiN8SrOnmDIcxfURi6jA+DnQr9kiLqvHXdOcG7Qg+UnWhzzIU4R3URVYdrAriuvYBRz+JIYjuEgJVIAz4K3dPG9imhvpVR2sRovN18+6bnfAX3mzwHpAdr2KK8K7KAE7bS+5wS4/94kRcB7wAkhZBRw2G7xU2Su/TwvzslFTKlYj3JmM86fpbiK7wHalCG8R7jnzlhbs8nMPFbD4VJ4Oky7gsNnA3tCB0mXw+ihRur/PtRI7c5Ms4Dx5ywx2iQj4s0zI8sB5CTivqdRLWjHG/kA/jAuPJNvBRQs4L97BxVbewa6RAiYPnM+NdCvPvrFlfO2swu2KxGfdA/t7tjKCXTPrga8HrawdC3Qh1mUK2DQbC1VW3fcG2YsKBsNRmVbi7qaFCO+03NMK2Guv5h3sCnKfKQEPVsCO9wpO516TfJCmEw+2JLmRmEdQy3DsLH6cn48WCRdE9u/GcJRuPkhrJ7xNC/+0HMU7K/csApYR7ArjPlMC9lbAQ0F8qwJ6oSQROxJmcURfiz3g4M8ftkiV87QtNbeffOqfQqN4Z+WehHf4WOQb7ArjPlMC9gZ9VGpram2S2Z2rl0tHYaKaNN55B7viuCdaWU/D6ZSD5NLs/IEk5gmcwE79MDrKOl7esoJdo7gLf7hpOZ3Sy5DaZt/dyp1mmFb1wemYMW0/rqKNm7eMYJcI9yStrqk6nRID+DVtZXnaBYx/WJCEx7h4ywh2iXAX/4B0uB9hAkECJsyugONOp0wa3iYQChVw3OmUacLbBEJhAhY5nVJ2eJtASCXgJKdTyg5vEwiJBZzydEop4W0CIbGA05xOKSu8TSBkEbDQ6ZQyw9sEQnIBJzydUmZ4m0BI7oETnE4pM7xNIKQSMELkdErD7M7LDG8TCKkFjBh1OqXs8DaBkEnABAIJmEAgARMI0fgfPJL+hQz6uU8AAAAASUVORK5CYII=)
Answer:![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 7.For each of the differential equations in question, find the general solution:
y log y dx – x dy = 0
Answer:y log y dx – x dy = 0
⇒ (y log y) dx = xdy
Separating variables,
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,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)
⇒ let logy = t
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ log x + log c = log t
Or,
⇒ log x + log c = log (log y)
⇒ log cx = log y
Or,
⇒ log y = cx
![](data:image/png;base64,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)
Question 8.For each of the differential equations in question, find the general solution:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,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)
Or,
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,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)
Let
be a constant,
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANMAAAAXCAMAAACmoIpJAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjoAOjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmY6ZmZmZrbbZrb/kDoAkDo6kGYAkGaQkLbbkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/9uQ/9u2//+2///bjNwS7wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB8ElEQVRYR+1W21bDIBCEqqVeaqJWTanaqMVbk8L/f50LSWwI4VJOHnpseWbYmd3ZXRA6ngPKQEnmkWrjkZEBQ2FiMYnUFI8M5RZ7j2U0UlM8MparjmOjZf9DxVS4NcUjh2FufaUkuiaeYnXmfLYSNHteD4T05GdIlTx9tNSJKWlnVk07IgfUVJ6/O1OQJwVoElSyLwnW9Tl5VEiUSyTo1zvPRPa8Vd5iPPJ2rPicYDzVRKjA1lNcrNUFQWVBune/ycnKBm2QKFc56VAzkaamkmRok/o0CTpeCxkDNV0h7eMQxW+WtRAm2bNxsK23SJkOlniBpqa842uoeH0Mxsa6Y1p920hBgQsUh82RRImFtaTdeC0kQC91axj6etny1J+I5iWPpnbAokkM6KFjVNgHQpdmCwnlddq7hhp1CtUkvu6IatdA78l4dROB+XKftzvKaiSf3dvb7g/i19TvPdlPykWt454RW008vZ5ZB4JlKav6gGN56m9EQ5N0RsBR/HVNvlm+dU6Ow/1dcVGeE7nsST/WnBEMZ0i8uaayDFKSBG0edvtMQ2dU7VAEGEjfEhIJm20EdcJeUT376YPg0xdvqV5hO/2knf3nRVUXiiArBD62B9egJxyDfA8IRlAQdPr0z8ok/3tX9g94RJKOkIPNwC9MxC+jcqMyDwAAAABJRU5ErkJggg==)
Or,
![](data:image/png;base64,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)
0r,
![](data:image/png;base64,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)
Question 9.For each of the differential equations in question, find the general solution:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJUAAAAXCAMAAAA1BP2uAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOjoAOjpmOjqQOmY6OmZmOmaQOma2OpCQOpDbZgAAZjoAZjo6ZjpmZmYAZrbbZrb/kDoAkDo6kGYAkLbbkNv/tmYAtmY6ttvbttv/tv//25A625Bm27Zm27aQ29uQ2////7Zm/9uQ/9u2//+2///bQvEOnQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB0klEQVRIS+1VQVvCMAxtpkhFFAURBjLUorJu/f9/z6Rraze20fHtsAM57JA2ea8vrx1j1ximAmo/Hh6xw+OiF1YpwLrP04lGVp2Q0ktYNRc1s2JdkCS/QKsyQD4HHdSohVUXpC57A0bdmZWcfFbaygXAlK9VDDA6Sg7R7gxutgGIZkdtEfocOMxKJSq5O+pEQi2FM61BcmmvKK2gSr5iaosAKh5hr+ryKUPap7a4Vwss+WTHRMmWSAPgpaCFYCo27nBILm2nTgVlWonmQm3FzVfFEnhWE15NPkdE+eBYaWoFiZOgEwi79o9USpsiEXnW1hj22CjY5tz8GNvD5INaGa2wWdGkLiSf2ul6SFjq0raqiZWKxywl5c7F9z3coqwhrJhwKvusTLpxgv5eHGFSeiFqJ0icfzi+kyGs8uUb+YLCR/LSxWLFziSQ8RXWPS9Nixa5yFKMPBLACg2Rz81D7yH5aY108jKI6J1lr8UlSszVaR0hXaWM/nT6iuhPk9tVgn5LbVOHVE7Xgqk9Pj6/XNNKrdpttLINB3giqfAh0h+yR53d8QmMUCu7aJFWNn3Ov2Y97eVfHwgWtA1nHfIsBPXqb5OKZ9vBSYU/BjLLNQaqwB/q+i7cmmppygAAAABJRU5ErkJggg==)
Integrating both sides,
![](data:image/png;base64,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)
Now to integrate
we have to multiply it by 1
because,
∵ {
- i)
So,
![](data:image/png;base64,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)
According to i)
Let u be
and v be 1
We can take the values of u and v from the formula (I.L.A.T.E)
![](data:image/png;base64,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)
So,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Or
![](data:image/png;base64,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)
![](data:image/png;base64,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)
putting the value of t,
![](data:image/png;base64,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)
Question 10.For each of the differential equations in question, find the general solution:
extan y dx + 1(1 – ex)sec2y dy = 0
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
Separating the variables,
![](data:image/png;base64,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)
Now Integrating both sides,
![](data:image/png;base64,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)
Let
&![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFkAAAAXCAMAAABXhxOBAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjoAOmaQOma2OpDbZgAAZjoAZjo6ZpC2ZpDbZrbbZrb/kDoAkGY6kLb/kNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29uQ2////7Zm/7aQ/9uQ/9u2//+2///bp53hgAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA2klEQVRIS+2T2w6CMBBEt1VhFQEFFMQiUgv9/z+0YDCk4dK+mcg8z5xMZluAVT+1wANpKtCp7EqVYboc4E7FPDtwHZDNfZkMbHs2cA0s4pAzI3Kxt90CYJYsAkLbsq+rTHb5UmtOaAoCSfwxzpEFRlBiDAW6MiPLrZkiAzchZwomE3eka0Z6tbBefIqs2xv/qDIt3lCTZD3f+F0xoxN3WQty23lUlmvojPGJZ5bROg9PoKU4iUAW5n9EoFfVYfc25AXVkNP/q0BCT8YHBLgp+/P7oA3vvtr+aoE3aTUOURD32hoAAAAASUVORK5CYII=)
Then,
![](data:image/png;base64,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)
Then,
![](data:image/png;base64,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)
Or,
⇒ log t = log u + log c
Substituting the values of t and u on above equation.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Or,
![](data:image/png;base64,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)
Question 11.For each of the differential equations in question, find a particular solution satisfying the given condition:
(x3 + x2 + x + 1) dy/dx = 2x2 + x, y = 1 when x = 0
Answer:
(![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,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)
Integrating it partially,
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAY0AAAAwCAMAAADaWtb8AAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABYAABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZpCQZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tpA6tpBmttv/tv/btv//25A627Zm27aQ29u22////7Zm/9uQ/9u2/9vb//+2///bSngzSwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFGUlEQVR4Xu1bC1PUMBBuToE7FQ84xCfWV0W04F2x/f8/zbyTJtu0acJdyTQzijNmN7v77W4e/ciyecwRmCOwlwj8vULobC8rORaZhBGHDgJev778nt0tfhzWkjFGNL9O/wRYXa5vpfQIVbp4gBWQaPXiwGgQozyNaPLzbVAYqpefufwoVUp8tBXNPW5Lx1boy4vRGkcJNvmJHclBRkjJJic2NzkiY3HmAQwJwWKNBQT8TNXgcf+Wxc8zewD9BbrOHjbPjQq/s7aNJhd5M9hIn4k7ZLdGywjQBim5Y07UmyXG5Abhv6EB6CjR+ba5oeuXLCe4KkAeEH+4QiJ8XNzH8fbcglhdonaoSwsMnHOPikbxYWWmo20EaIOQbHIWf/az3gC1xv7b9KNaEYHmC0GDZbdQNQiNan1bCjTCi4PmAkajQOhkWyKCyw6DsTOMNr3Qpo9PBSlZv/lNW5XbCAgNIZlVK2axQGOpK1Mm2jpKrSpZi+KqIBfBhJBoeHa4jsAVBNwCW0UXqzek8/agoaZHACMrlxlzyWkEFAopueNBpWj8+3qse6SZaOlocr1N01YhVEljXGCS1iI1UPHAwXKBbIfd+ybghWu6n0VE+QAjADSUpEKD5NIJOaxCHlk62j2tIBWqVFkuumsjo+JhQ9RXtXqNWxQwcMnyoW+1ndP9ralOt6JZd2mFbciUpEhQWhsNSWvScloegToANNQ+0JLvMKFVG8FoqD1L76BmSKGzSLQbIt6u8GD17jACPA8JyVanEru4rayvU+m1ARjz6LVRLHl11e/emyddV8N0Tveqj/qSJDI5SuBNy2GEHQpNso0G2w0AZbYOVkZ8tPYNQL4PjdB9g56Rd/gogc949KwODzunnNO90NjR+mYXBZdWOxSapHmmwjohZYAORG+N9IRbb7QzFSTvRoOJBwx6RG6KC/oH52eXOtOMnuk+FvFWWeM7qFsr0GVo9hBJeUlgoBbYEVCZHc4mX1zj6yLbaPT7BiQPoyGqK/i+wbemC/yisMC1gbrgMMzom+6BBlkUJzL58ezKaYQZCk0Sw8Eu0Pxl5AgHGPQIOpf9XCF09JHUp34XB+UBML9haXT0ie55wZu4R9ymPTXC1WvkO5WIS3BpTDvAftaNenjVlzj8G66fw9OePeKjhO7QdL9vTDvss3VzBHwjUL1S3xJ9Zef50SMgLqTRFc8KB0eAPYGzEe3taPDqHhM73908dDypqeVC+zChQDrov3oCeFDboi/ecraFxpPKo1SMnTvVNJGcd/EJ4QKdcEdcUAMvpTIgkdl3CXgy6vGm88HGkxwbgX2ncj0BTzwfNk3WnCE+mBwbj30n0UjBk27WHMDzsllzgLj5bAx8MIjJvpNoJOCJgzVno2Gz5iBxk+RjoxGffYchScETB2sO5HIarDlAfBg5NjL7DqORgicO1pwbjayDdDeQHBuXfUe6VQqeOFhzfWiApDuAodvHaYn0qTgFTzpYc4QwDL4yajlNomiKmwzdvbDv2D6egifqbj6Adaf7zBilLnFx2NlzbUBkRJAtrZjKk/FEhnMQ685iW7vEh6IRyr7j66TgiSTgARQ+Z0b1ku60WxnwazayrwSz78Q6KXjCT+kDWXdaF+gl3fWgEY19p0qQFtnT9oTeYAey7hqbNecQ7+5UgJ4IL6speOL5uiP7Av8F5ZHiMvgR2XcjTQmkD0b2JO7Lp2eOT/gN90CeJPBVgEcuHU88U2GePkcgiQj8ByTv+AOtnhkPAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now comparing the coefficients of ![](data:image/png;base64,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)
⇒ A + B = 2
⇒ B + C = 1
⇒ A + C = 0
Solving them we will get the values of A,B,C
![](data:image/png;base64,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)
Putting the values of A,B,C in i)
![](data:image/png;base64,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)
So,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Then, 2xdx = dt
![](data:image/png;base64,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)
![](data:image/png;base64,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)
or,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Or,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, we are given that y = 1 when x = 0
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALYAAAAqCAMAAAApilEOAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjqQOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZmZmZmaQZpDbZrbbZrb/kDoAkGY6kLbbkNv/tmYAtmZmttv/tv//25A625Bm27aQ29u22////7Zm/9uQ/9u2//+2///bck2NvAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACJElEQVRYR+2Y6XKCQBCEWRMj0UjMZTASQ4wc+/4vmD1Ajl2opcfCVIr5oWWVH9M0M7OjnjfFX3Xg+LSnSCPiYOpsw26+QFZgRBxNnK4OMUE2EUdVS44im47DyifZsHUAOLkNmAYjk9uwdQAYz0inJBEHBAuE73zG2O0bRlNxMOuEXcqBy2xmfMfY8oBqGkxfaDPj4d0pC9Fdy07z8L3LBnAzy5/1DpsV714ip0zCOvM087dxO90jG9zMeDiXuvPgodATSaPzYOFWJW3cTltkpxvGZtob6PCTj7UmU30Un+WrSzTxDtqUnfpbLwvssiMxn3X0HS8i1c9mUarUifWrUzTwDtqUHdWu7+T2+VZYqYqHrLpIK7Fx5/24RbbVuzxYV544yTYttMh2rm156tbuuiySVmcYbvfKHlAkld1FU9Xc6K8VVSRn3E4Pk+1am6oly1qO5exzHoC6CyrcThuyeVh7HshmVjRfVRVyiLh3ZBu302ZLxmzr8U9xQoCLXfaqbU6Ld/U/CFu5zhEDt9KWuf3ts/mHUznUvoQ8lwLnR3Fbc9K+PlRt+f3Ux38nROLxZgG6s6CKJZcHLwTZsplU540d0VptPIRwnzSEJC00WZ6ostV4Hjfyx73eTvFI/dFrhIfiABSyYzyzusTIkZTLECy7ccSNqp5UJNF5ERhVs0hGkR3LvSkZv0zk3MVbMr2Xm0R0DdmivmHdxTZ8Ddlj1+R/y/cLu243FLncmdwAAAAASUVORK5CYII=)
So,
C = 1
Putting the value of c in ii)
![](data:image/png;base64,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)
Question 12.For each of the differential equations in question, find a particular solution satisfying the given condition:
x(x2 – 1) dy/dx = 1; y = 0 when x = 0
Answer:![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,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)
Or,
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,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)
Now let,
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAi0AAAAvCAMAAAAYYApiAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZpCQZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kNv/tmYAtmY6tpA6tpBmttv/tv/btv//25A627aQ2////7Zm/9uQ/9u2/9vb//+2///b1glsCQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAGP0lEQVR4Xu1c6ZrbNgyUnO3GTdNN4jRH29Q5nFNZu7Xe/+HKm5QJEQBF2dJa+rHHZ4IDDMFDJMdVtTwLAwsDCwMLAz0MtF+f/iCS09x9cyUZZs4mtCdCdmAy8fmwJVBtgHx0IjWXKNZun9+TcQ9P/jFlWWaufm9PhDyBycTnwpZBtSFy0YnUmGI/X3/iGWSUbre3Okfa7QuKufXp8Kv2jWjmqj6178dst7V8Vs+kfxEME58OK8B+vhKwdwJ2KKoNjoVOaYW4zL+v6kfUmSEPQVrt65Vu9j0FLPCp0VmWMGu3dvxx7sX23vOo+HHzWDTYl1r8BGBwfF81C7Zq6uf37RdJSxwcisoMOr/hupaHu28NpQEH4u3+XKsxpd3KRkGe0CfduVNmMXGxfQAYFdd1HzciLQEYHN/VzYM9rCVs+/5TFiozaIxxxudnyJbjH9/1VHRYq4FgV9e3901dR6OCddv5pEfplBnQzarqxJ6SLaLxAJgQH/WaA9uYwRZFBcliBs3IBqzoGbKleWyI3BuOduI3GLFx1vu0UzNEwixNXKXs0Wz578MvYjqGYEJ8zGufLShsu3UjOoZaAbDMoLEUYHw+frbI2HS/tczIVW+TWO8G2SLHpJQZRpxZXltCgJlIrnJv5bs6BLML8DGvw2xBYNXMpx8MVXSriCxm0IxswIqOny2Hp3ZN4LAO69+fGcfEAG8eNzif0g6ZmRktMlW1nmRbAkkvG1rZf71V6J1sVVsd4jUDNsgWkJMQVfQ0R5aexjlBY83P/Hz8bBErFPHIodf2I9EAPjUAf3vGFsiM2c1Sq1zIu3BswbxmjC3ATBTW3kWNyWIGzcyIVPHRs+X4Ur7X7OWa1rXH8c3b1KtYz7qlAsww4kjrFjHCCHcg7zrrJsRrxrpFLUa6M1EYXAcViJoZ9JyyZa+maLWvYd465Juj+r/vcdly3ATvRKBZmjhtHzypsQXwroOPee2zBYfd19Ix+QaNoUJRM4MumS3JOWE4kNnFOG5E7zV/tztBleYLftw81dnvAM1g4mxMdjPWwfTszu2kM4B3IT7qtZ8xcNh2u3onNgXVu6HqNmHtaNTMoIc3oq6h/bgWS4qbv0vVF9dz3MjNFTG0qJWL2rcUu+0rMbbUPekS+hTuasJmMXGAvXfrtLjZ+b95J4vE3gX4iNcdKo1ZP6wg4bNg/uYv+cYXcYJGzQx6vNYdt2bmgQ/znCbyPerj6ehKndjwYEuh2th46OO298DaWYfJmWfAzkX2cWyZ02AubBlUlyzu4H5gU03BnHFRJfN+iYsy46pHiZsmfNgSqDZqPvoUsmLxYWFgYWB2DBx+8zcQZ+f84vC5GfBb1OdGXvDmwoDcoeg5UJpGCL3nXdNw73q9aFbBTSKfRBf5C2mFi/h09aCdRulky/V2miXyXgaWmWhJjhwGllVuDmtXapN8g2bslV7X9l8GMVXmbvJs9lNZ5zDA0QKD09lwosLMIqbKPKninhRdaNhjnvFG8jYap8ZsJpyYZCGJJF2zZaolz6AXLJhZ/aI+kryNlGxekzejs/GhasdMkWXBpi1fVULUR5K3UTgJNXnRrZ/yIZWpcajaMVNkWcb5srV4EV1C1EeRtxlOUFGeFzUYVXvZeIrVRiIGlsEx1JKhu/0iyWJBDa7IqdkSoj7KNXF7oRgT5c2BE0UqhRgkW7TsMFNkObhlx6jAqdkSoj4sWzBNINiDIgXnGNENqJNCDJotmcycqBUHRFHa1KrZekR9VHmb2/Xz6jjQFJYClg6qRH0IMWThH0DsjJkxkoqEFpAytvgsYEkJSzTrWHXgxLDGFkyuOIt+ZEV0CS0gli3h7AxpAntmoumOt8phAjFotmQyQ/immrG6SLpeJ6JLaAEp8ja37KFKCSe/brGBpESSFGYyRZaXSQckWZz0L6EFpMjbDCeoKM/NU5GCc1r0+EBSIkkKM5kiy2nRobwJRXT9WkCKvE1zwpASTnszl0YMMBPx1JIuJZIiyQkmDvvLH00MttUpe7mdBczEly3eV9KZBtCimWrJaXcjn9+c76O1Vu54kMnpTDgxIzDji3odnQ/7DFrMI/TvuracBFcPaGfQdkSalYYyg5gHf79l6PTI4HRe91uGErPYLww8WAb+B74BHRmlEk7XAAAAAElFTkSuQmCC)
Or
![](data:image/png;base64,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)
Now comparing the values of A,B,C
A + B + C = 0
B-C = 0
A = -1
Solving these we will get that ![](data:image/png;base64,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)
Now putting the values of A,B,C in ii)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAaEAAAAuCAMAAAC251ZJAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6Ojo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGY6kGZmkNv/tmYAtmY6tpBmttvbttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///bV8r1ngAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAE9klEQVR4Xu1bfX+UMAyGqdv5rjdfTzfUyXbs+P6fTyh9A9L0SbndBnJ/+FNs8iR5mrSkJcvW3xqBNQJPMQK3n65PbhaICQ7LMngg4ymkAxp03HDeX+bP/h5XZVQbiAkOyzJ4IGMZpAMaFHVfNqB6d1OemiEQczysvrqhvOP0Ve+xeEBgISAUBDNlPOrkDDUmgJiDYb8+hnwM6wvLDHVBYDQQDpJEEhitJN3yiPYk+qaVF0Ebwj7Uu++g5RAYDYSDgLb0h82FoepleEfD+LBHi3hPRwgsAASDLJqhIpxCXNWsd8HayMzUEFiAIRhkyQxxKcSua+j89oMfBAslKwqyZIbK87uwe1ylPmyxJPJ1BMFCQCjIghniQ8CupQXHrQuZpyMMFgQCQdIYOjt9T6HEML1he1aC1ceL2phBYEEgEETOUP1zk+f5869yyXQJELM/jMmSmL5qEy9zEBgHhICkh2wGkvUOK1WUK/WO2QXSAnIwMcgMgi4y8bAVRtnXLl0jksCkICL3ZzB4UhEp0JdWHYgkMCnIDII+MJHvnOxztHlDeD5a3iNdmiQwcPMzP2KsxYtiqHpNduFnTE9jOs/QpBk6yolIDiWB9UEebPP9iCQvgaHDtnl10T/sje/EAS+mmLcEhvx4l2feyumIe5S/xebBgDhnoxWkqO0PM4VH6qCC6BUgchpNAxuDNE96DMVCNIv/X0IOrVUucaqtO4XEwPXF/p/ddv37jfR2VfnuCezeF8UQN2fr3QfmpCsgWb2y+w4BwccllmcoqRFjvB01ZCLvQ0lgaNdHeGBurlDaU1+MYC3mEXuUQjcoe7eXef5Cn13Jm5ne7VCgqVmngclAuj1fsElIzBvvCqU+9oUIdmLs3YGppBX5t+x+q/3hTx/Gvvm3Q5GDAQGY80sK0koyxoy98K9Q6lgjFyJ8MfbywESK1HWb0nRM2XPukW+926FI0RKAWbcEIM1L2PldmTfOVBu1oLgHLkpk7bV+d8mjCabEe+G2YilJFNXuQ9mNsm500cKUb45SuEeGgfnmwSBF0wRSVhpj7AOUoUxNIk1wRojTDEFVcZhLMe3+eLv+mssdpDDPELAMdZAYGM1QBKSt0mWbBIYh+wBnqD34ZcRphjpihT/CuJAGM2XaetedTJPCLEPwRSkQjGQoClJt3qpL/q786Ad6coRami5L1Ryw9cDoUwVz3KwdiAkZalK1szb681PUFNShcKhda21E1taOe3fhhADjAxEH0Z1FV3Kxc8VBqN0/+SOSiQw1XUXoAlh/32PSlRDmcggtxDAYlUNxkMPnL2pbahkyD/Aq15YrRjxU5eS3YrJsbBydTcWF//Kt5zUlzDEUn926zqBgFENRkPrHdfdKZ0qpfYAy1NVRuw4ZfYE65OVQyjoU0W5A1cqzd6VHfZ1CeEafz2ob0Q9HcDCCoShIXTRu7PPmD52q7kGMIVNuugmqCabE+zlkxKILJEFxVLuWUSapwebXfEhFC5M51NkIfnwlAOsxhIHUu/ysyaG8pUilm//A6ht74V+h7HZKHcGkuEe1d0014X0oot3h6KXZY6i++uO56kVq5Jt1Df2AUQDmImoCgYIoyfiSRRYuE+loPR1IP2RPIVBhZ/8Ya30O3LQtUCHBCSk0+wBPd0BwfGCXZHc+JCL4YXvb00OxTA0Cgo97PrTMcK5erRFYIyCIwD8meaLmJw/hRQAAAABJRU5ErkJggg==)
Now integrating it,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now we are given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Or,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now putting the value of ![](data:image/png;base64,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)
Then,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 13.For each of the differential equations in question, find a particular solution satisfying the given condition:
cos dy/dx = a; y = 2 when x = 0
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now y = 2 when x = 0
⇒ 2 = 0 + c
⇒ c = 2
Putting the value of ini)
![](data:image/png;base64,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)
Question 14.For each of the differential equations in question, find a particular solution satisfying the given condition:
dy/dx = y tan x; y = 1 when x = 0
Answer:![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,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)
⇒ log y = -log (cos x) + log c
Or,
⇒ log y = log (sec x) + log c
⇒ log y = log c (sec x)
⇒ y = c (sec x) -i)
Now we are given that y = 1 when x = 0
⇒ 1 = c (sec 0)
⇒ 1 = c × 1
⇒ c = 1
Putting the value of c in i)
⇒ y = sec x
Question 15.Find the equation of a curve passing through the point (0, 0) and whose differential equation is y′ = ex sin x
Answer:To find the equation of a curve that passes through point(0,0) and has differential equation ![](data:image/png;base64,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)
So, we need to find the general solution of the given differential equation and the put the given point in to find the value of constant.
![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,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)
{Using the formula,
)
Now let ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
integrating for
.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Or,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMEAAAArCAMAAAD2QxxKAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpC2OpDbZgAAZjoAZjpmZmYAZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkLbbkLb/kNv/tmYAtmY6tmZmtpBmttvbttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///bntfDDQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC10lEQVRoQ+1Z+3+aMBBP2FTW6epG92jpVtgzVhma//+P290lASMtMsyh7LP8AIrc43vvRCH+r3/TAvr74hc7MnX9k02GTt9uauaFvOORVL5mYix0+m5fZTYEonyV8dimeMEfQkZzNd3zdTgwOp0Rs+29lNFyU0iIIrysYrnE57mU043Cpy2rvAFieAPvC4x3y80nZ3JCGZNuOp1u9GcwEn0v43kmlNE6jzL4tR1AfCu2yR2QwT0Fn1bcPPKDeA3lhQIUhLVLIBvKRYWAYFCCoDbKS5WG6NyGR47uRBNU3Hxy+j34sgjEVzn/gcytDwABqUFP3lA8uQVxZZcBX71oCOjquPnkDmlYEMol8uOVnEBONxEIZTV9TrCDau4msSw3JKnJeRA4H4CodQyymwh27z+0lysfgftG3NAnNTkrAkwBgRIaCPR9tktaA9iVM8wYY4KKG7hkj5wnD2wtojJyA4pSR6OLyWSdw7WQrams5K3Q3zJ6bYswKm4eufNO2DSwYQsFPJbyGmVD6afLLpGgkE5lBD7Ajy0LmsfkC0Y/EC7BD46bT87UD8Q4e/JjHGVlbKo4U59peuyIC8Db8i/6RQFdCt1tIOzPpoGDtGZ3dDY9Ui8OFVMvP1WPLmR/4EpaRxuurngGxY7isVfPD7Y83RDA8Bih8X8/6HTCt2nqhGOvl9po7pAGWKfX0HFW8UznMDV3khT2JazMB3OVFeD7oDF9mbew6XZzVli1n+GmcE9Rp6PdqbSJNk3Rm05qgwzzyVOvDwJSc7Bt5VPWPC2KmAaTnhHXJ5MvKAWeqKadOlqBQ+Sq7mU9rcdC1nGqgCEy+niOIsoCujdTvYZDlUnWm/78hDnE4jY5a0U70Qi0WbSnSSeyOic53/nqUKjyMUcRGskeEAxlr/ByBtuahlfdcLyo5t4LZD4beVekvzKK9vPsXpYZioiOTegsb6zLbgBHjGCslg+j9x+W10iMjC16pQAAAABJRU5ErkJggg==)
Or,
![](data:image/png;base64,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)
Now we are given that the curve passes through point(0,0)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEoAAAAqCAMAAAAj+j0ZAAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOmaQOpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkGY6kLbbkNv/tmYAtmZmttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///b9yTx2AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA7klEQVRIS+2V3RKCIBCFWcukHyyzIiMT5f3fMZxxJkFpkLYbR66Xj3OWs0DIbFd5vOF4q1NYPVBQcl8IJJTWs6D8L2X+vYpw0q4uFADWJ//WLpX/6YDcFRZYpgDROeS0ykqHpBmp2QRUw3QqumWy+OZlKOKOulHZom+nYQcfbx8hRnUIqgdwGhyommDQarvKEx+DYzWDMAjIiLrjzPyTQnwNVRayT5U6yTGKeK77UDOUT5W3tyNgwlR8917hoTiKwVaupFj+VO41rh75+GEubDpPzNfI43RHiWjftQrDotzqpCuOgereHgxUeGfcO982OgvMpjQZNwAAAABJRU5ErkJggg==)
Putting the value of C in ii)
So,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
So the required equations become,
![](data:image/png;base64,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)
Question 16.For the differential equation
find the solution curve passing through the point (1, –1).
Answer:For this question, we need to find the particular solution at point(1,-1) for the given differential equation.
Given differential equation is
![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,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)
Or,
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now separating like terms on each side,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Or,
![](data:image/png;base64,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)
Now we are given that, the curve passes through (1, -1)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ -2-c = log (1)
⇒ c = -2 + 0 (∵ log(1) = 0)
So c = -2
Putting the value of c in
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 17.Find the equation of a curve passing through the point (0, –2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.
Answer:We know that slope of a tangent is =![](data:image/png;base64,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)
So we are given that the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.
![](data:image/png;base64,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)
Now separating variables,
⇒ ydy = xdx
Integrating both sides,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now the curve passes through (0, -2).
∴ 4-0 = 2c
⇒ c = 2
Putting the value of c in i)
![](data:image/png;base64,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)
Question 18.At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, –3). Find the equation of the curve given that it passes through (–2, 1).
Answer:We know that (x,y) is the point of contact of curve and its tangent.
slope(m1)for line joining (x,y) and (-4,-3) is![](data:image/png;base64,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)
Also we know that slope of tangent of a curve is
.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARMAAAArCAMAAABsKZz+AAAAAXNSR0IArs4c6QAAALFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZjoAZjo6ZjpmZmYAZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLyQkLbbkNvbkNv/tmYAtmY6tmZmtpA6tpBmttvbttv/tv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/7aQ/9uQ/9u2//+2///bH8abzgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEfUlEQVRoQ+1ZbV/aMBBPcEo3UdkA3QNUN2dxmxbY6Ar5/h9sd5fH0tSWUpn+aF5UbJN7+OfucndhrB0tAgeNQML55KAB8CmftJjkYEmD1k42QWkxySKSjjg/DyYi5PxkmQa8c3fwoSUNxkzcQDwR4cmSsaSFhLGIkMAYGx89wqN78FbC1sMBgEDxBB/iuvUcFxMRdlmCVnPow7ETdJ6oPZMZRFYMIDJnWw/fX0FIaUfcuWWrS5nHRhyDSzvElHf6vwMCJcGTZ79D/Dwr5hlfPOxXGg+3pKmDOK6a+Ynww1NBPT31xTfxle/HoOEU3uIgLtvBiqmfCH3OKhaQWh9TUpC+zecGGALjvTi5CPs3pWZCsZge/h00tlcRE7+zRnzMVkPpx3E+OTDFmQ+wUl/bZhHUOxeluYnFxLuDVqBqmMhDLzcifBvLsO9RwTQ2qjHZIF9rUSnSaoJnBxlbXXMI2UtVNmF5eQZhMuGd7z8C3iclR7zzRZFQO44trRl+pocaSnPrXYZYwGULbAp/+dGjmL/juMxQgQRjinOgpNXc7De1qKqS5fOoH6ebcnoHRWjLRqwjxQ3IQruB5eUqRB9IeO+BzTEw4ruF7tboPUuD3h0YRu+WxYZWpM5AMhrE0iGmAi8tB1tbikiyIyrwMTq6YzPycM3NfmvcTkxdBELqHXQxoaw4PVOYkDq0hDadCm8sOY3LWExIfoJM6Wt+UI0KY4MYvrLqWbmQCtFfD+FhuKkJwCGLSYT2RKN2P8DFREq5Maa890uLK8sGekpHAAnlO60nVeIatwxt6zJqbo5YASY4T2EyUMUcUiDiUpRy5Q1O9odRMw9iFhNfvQjOfYzOApylGiSgxAQgWA+Jj8Jiw05Uba4XSTkymDjEDCZicQnRQ68lpuhKc3hluW2FSX6rn3qTxyTGILEaOvAsAgTBYOKzE82iEJOoa45AZY2FdoLxxPY65EZgz7AzVoZhLdFjJ437DgKRxQRDCe0saisbd2486dpQYiX1+A4daQmlZRILFYwcYtpBdTTP2InO6OxZvzc7yccTOhxG2lkSsKGV7GjiuThDT04gHxMzdRhrmWUWKGMsPjB5FRGBYfITl5iJwwN2/w1WrD6ZQC5jrM5yDbcsh/smG2dOzmZ20PW11TW4NqR+cF4jAHP4D5IVROI04G9ucSrkIJ3P2jNkHguNcT6xDyjPaZBiNgvSxIC2vo2C8nWCaUr/z1ARwAgyAOPFgZwVNyIuv8F8WFQ+qt4DOnL7S5EiVkV3av56x6GyTSZulq1HqPTfHe+s6twDevPYAlAK6deqi8s2WR7xOhKVzS76XuPOa6sdLMb8OfonGHogv9+xGbQ9JiV1cRZ88rm6+1VjHVZAfJd+lLoHxMh2soToVEn4sv5JDUVe0BJzD0jZnwgrQfKC5H8OUew9IOZY++lfPYceDdJ073fS4LzfIOlXS8rFxOlevFp9mhDcxWR99XHH86sJif4/DeceENr21Ihph7kHpKKL0p2DH/oecBxC2abKpIMHpQWgWQT+AT+xnO1oAQzjAAAAAElFTkSuQmCC)
Now, according to the question,
(m2) = 2(m1)
![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJgAAAAuCAMAAADjjyB7AAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmZmZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A627Zm27aQ29uQ2////7Zm/9uQ/9u2//+2///bWhSKCAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACcklEQVRYR+2YaVPCMBCGE1TiidQT6xU1Kj3y//+em/Sg0LVJN3GmzpAPDkj68O7BNm8Z268/zEDG+UNsfBxmFkOY/rrm/PC1iTAKMxcRMib5ihXJwUetLAozCkTOQZJqcx/MzKEA5+JBp5wfrXPBZ20xKG1XFbBmMmmYitbBuVgx/QQ4nR6tGcvCdDFpStkymQScTkl9Iq0cE6cyTGXqQV9V/TZME61aUnhlYi6zPPNHPwYVUqeG1mEC9HxB0dWF6HTOMpM/8jKEHWFMEXujGx3UUpLaoYlEzm1YXWZ5e9cOkFEBV0FWv6UyubxtptAoSL1Z2XZdQre3TGiNMqG1rZq9sOKmmj6Sk/q0majH0J9aAqJl2ncZjarf+GzxLayyjJb1WhgMLbNASsNcpTAVy8T+L2hltKQHfafzYuiGwFnh/AraBp0unqaYMPgp8YuQGUZLxySvyk/fJ6kr/MAQOSwzPppFvF3tKqoHFVAjAWESd26BG71+rxwJ84Nsdm3htoRFrsxo3D8oZehpeXROPC+Y7rjwDOC/bNOfJ11fHUM26ZTd81jSHCBT6sDCLFsuKLS+sG1fPTJjiLAyuR8U9ovjxl2pta/DCzfbCE4uHeMBd9yYsOL5wuP+j5rtPi47W7vmFuq4+yQzoX1OaKjZ7uHKq1fn44ee4/715vwl3KU0Dn7bbGM4a9EhY2rIt6KOG+8xP+eFjIFdHDxfrNagocYcNy4sFx7GCzPbKM7VY6jj3iVVj0d8HleiZpsmDHPcfWEwdAr7yGx44WYbFeae/Ijj7pGKZ8H5wq0LN9uYMOgz1+jfO25XH3Q/3zvuMdmawt4fcEE+yMhiFXgAAAAASUVORK5CYII=)
Integrating both sides,
![](data:image/png;base64,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)
⇒ log(y + 3) = 2log(x + 4) + log c
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPgAAAAXCAMAAAARHXJuAAAAAXNSR0IArs4c6QAAAJ9QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjpmZmYAZmZmZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmtpBmttv/tv/btv//25A625Bm27Zm27aQ29uQ29u22////7Zm/7aQ/9uQ/9u2//+2///bCd4DdAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADJUlEQVRYR+1YaXPbIBAVzqEebpxaSVpHaWzSi7itVVn8/9/W3QUkQCCDPnbMZDyKxfLe2wtwUZzH2QP/twf+3DO2milRflv+zDUVN99zTeLzZxAw+N3dttgvdhNcBIu9lvXHQ76I9t1jglEc1TKeRcDCb99MCS+aiHBZrxMU6Cly/5axS8I5AacNYqiO7gwCYCeUkAFfTNvHKDQX0TyXtR9VvtgWx1ohi6uEREkQnkUABJc6ggZ/f6LEIxRkfR0NeEA4ThaM/JEU8tPC8wgUXfVZC9f4wtPdvve6j6LQQhNc4hv5UjIYV79LVCFreDy0pdsIxsLJR1zlSFKNRFAxV5DK4hEiSG7kSEAw5VMzAgT42vhS4Tegu3GMfF/T/225gVRF4vxiV7wCjJ4ma0xczyYo/PhsGjqP50pPPYKKupFK1RMoOPDzAccEmuWhJ4n4XYXhA+HqQQ23m9F84oo+pgTrquteq0BvCFdKQDiuf6Nrm9tFDhELwUZQMcTaegggRNxrUyMC3d1uiI6DbyeKgEQaBgJ0Fa6Mn1r4uiDBxhtPWA1qRITAm1+lNokCn0ZVJGgYAhCPD3a5hghQdoMQoaoj1l1jwkk0ZtZ+SHXyROOtFKnxhinOecIdVEv4UF56n7Lc5hNoTFaNhU+n+hBxameLjVXW4Hnu7V4R4W2phduFMZHqAdSQ8O7hk7+zBgk4NW5nuH4ONTfVwqiPmnLSTRWZ3D54O/oIVzdS1Xr7ZA2A918hiwAq7gnabYaAfNphz3HGpPAI/ng7Q7qYpUdU3+9FwzbKdQYP0AHhoIPssStMnxTVMk0YFWubbQr5Fd1CaiXH8vUoBIWbikjCxzWpy+9h914hb9gyYcBjf3Aan6DGuMdnY592couiAoPXkl1u0TeY4FB5C4g4c5WHhOsl0/DHydjdY7r+pY1Nt9cmYVu2Fkp0uAM9oDodLO+srk3n4GPAqY9QmeDlCAoM/qaK1X+XdjvzrCxUR/mM6+EsfKo9lPxC8uE6/KNefckK+Lz7uI1qK8//QWAePmLi/ZL1vyZAiZnzWE7Ys+e6qNnmZ4OzB4wH/gHweGS7npsFogAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Now, this equation passes thorough the point (-2,1).
![](data:image/png;base64,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)
⇒ 4 = 4c
⇒ c = 1
Substitute the value of c in iii)
![](data:image/png;base64,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)
Question 19.The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.
Answer:Let the rate of change of the volume of the balloon be k. (k is a constant)
![](data:image/png;base64,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)
Or,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALEAAAArCAMAAAAACpnSAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OgBmOjpmOjqQOmY6Oma2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkGYAkGY6kLbbkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/7aQ/9uQ/9u2//+2///bihm/zQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAClElEQVRYR+2YXVecMBCGE2yNtXXRVqV1sW1cWzEl///vdSYsMAkBsmvG44U5XngWeObNZL5AiPf1Zj1g76Xc8KljwNcbAX9sKz/efPrNphbADPj64xOn4vx4WzFGhBAMeKNYFTPgG3nLGRQMeM2rmAGviy2njxnwNa/i/HhbnUTK8d8rKS9yuD6OfxE5imwv78Qui+8ZFLdlzMfYq876+G5kcnJqGexzik+nzaRAW860PD2W6QMqVDNRPMGn0oyKn/Kc4h0JY6OSS3aCYkcbT3AuptvyW1yxUVEfa5p3L1AcwTtasDNz/itUXm/cPfaH3K/+zD0kxtgDHkYDghvnWANV4wvYGC4t5zla0VJ+6AUEinsaTORSnhBUeDai+fzkfrvf4vBHfdbIU/KgUec3j2fbtkQiKjbqRtjv+O/+0kphQiu2Oh2mwRDf0zqFnZ1u+SHQXoLbup/WFAdjkZsVXa4EI00dN4VWwD3DvnzFI23iU6ELkixu4oN7NBpe8bGfY22JO+hyJSn9wMqOCIbNkiMktBXFEIPdegXFYIWm8rLi+ajAQ5qJCu29lIaOnPfxfFQYRdzq4Rd8PPX5oLiBFkd1LSuGJMLNdnGcUpRdHJMW6eEJbb26CYhsl3naKd489y3YVzzpR7q4E/++ooS0VuXuIvOajx9pmMc/Vwrlvn7UoNhWxXV/tzdxGxV+uYCPDcXFHyVvp5di9iBjiq2tRoo/0A80qLOS1oZDhrk6eco5hDrcy4DPP3F7O2PAMyCp5Px4hombCmbA24r1ixADvi3pIHRUci09xIDv2hDbYsCndbKjd8SAx6bNuBjwmjXxRH68rVLGm6MPITce3kwaRhcz4GGeZhSML3Gc+KMP/v3Bt+2B/4v0QSgGpTuvAAAAAElFTkSuQmCC)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALgAAAAtCAMAAAAqRlEFAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZmZmZpDbZrbbZrb/kDoAkDo6gVg6kGY6kLbbkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///bwUlc4gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACVUlEQVRoQ+2Y6VLCMBSFExSpomLrAlUgqGGpKXn/x/PedKF7SyfFZMb8YoZO+uUu556UkP9lRwS2Uzp6swM1RynfV2Q3WltIDsji1mBwXh/VcDk3N+DCqQVn9Npc8KP32lDHe2dhasiZGyC4XNJ4uQGliy2k4fi8JsJY8ODxoMA3a3H3HXMK52G+h67cOubKIUZVgYOAZMBdU+sj4ZI+IAI4h0rOghtb2Ak4lHO0bANXB6goFeMjngUPrk7NaQd4PDm5And/oFuxcsxfUOdKVhiASx/soHAoNV5WzI/rcIQ8SpjOxWdfOrer2ytWJU2vUg0m7i/RZSl42bxL/2yASBkucg9Iwcuh7wEeJ47fHLIpFA8DFA/yckrHHziCQZoyqwJcvICK1ecBralyToWrl956jBBxT+lPDskIbgQXzpyEXkMBxUZauShy9BJPEmt2z0Zi6TZZIUFwcMepd2gEZ/kSKIMkNwA2KfzHs3k6nafhV9spAXyH3IWIV57y6BVGW+mpFLx4whx4G1On/9Fiqrd0aM4SeOkNOfDBS0U4mFj94KdjDdWcSn07gEMTt2SxusaHkUOEZtCh4C3JZ278l+WQ0zmR+YfyJ4mtaXtNdSrjpoeUu5Q+OspNUaErdBxu4ONV/X5oTc8enXI3hTmi2zH1i0xhcjZvwkYrEvravV4f8vO8itJ8bsL1p487xIvQX68efjxcXsbDaw4NDqpZzk5qfsGA2+0dA0ql1/kCW2/4OP4sW5F3t+RrUDa28DVoDQOozekbmI5wCUbhyVJVMTCeFiL9AntOM4vjENmNAAAAAElFTkSuQmCC)
Now, given that
At t = 0, r = 3:
⇒ 4π × 33 = 3(k×0 + c)
⇒ 108π = 3c
⇒ c = 36π
At t = 3, r = 6:
![](data:image/png;base64,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)
⇒ k = 84π
Substituting the values of k and c in i)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
So the radius of balloon after t seconds is ![](data:image/png;base64,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)
Question 20.In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 double itself in 10 years (loge2 = 0.6931).
Answer:let t be time
p be principal
r be rate of interest
according the information principal increases at the rate of r% per year.
![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Given that t = 0, p = 100.
![](data:image/png;base64,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)
Now, if t = 10, then p = 2×100 = 200
So,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFoAAAAfCAMAAABQ4yrvAAAAAXNSR0IArs4c6QAAAHJQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjpmOjqQOmZmOpDbZgAAZgA6ZjoAZpC2ZpDbZrbbZrb/kDoAkGY6kNv/tmYAtmY6tpBmttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///b7mQvcAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABH0lEQVRIS+1V0XLCIBDkaG2wtWlstKKWKDT8/y/KXaIFaiQ+4Ewd75G529vd20kYu+PSMNmIyT6LQsVX5i0TdCbKzgf1H6GNgCLLDR+gt3IAwrrV2jN77K4CeF7lYCChZj/l0/c4bLumUNsv4J/JCYm9ChZeo3ndRmOm6qCaWUXQqtib6TilOoBmmodjRtRsJ2i7Qui2XDC7fE/SxgaJhrTlbwBCbOm+SHbZ80XoD7ea5PYlT5MRKWY6Rl4p7r20JRJE/CPrGHqQ/xltETSRolNfZ0gvddiQjjWVlS/I3SkIzjhkiCyw26/ojMfdmCRXbk/rElOnr0h/De2f+0/4tMu+bdJBjpaRMCsvJ6kRwOextiTr3qZxIU2iPRru2IEDA9YQ+6peKQAAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
So r is 6.93%.
Question 21.In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).
Answer:Let p and t be principal and time respectively.
Given that principal increases continuously at rate of 5% per year.
![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
When t = 0, p = 1000
![](data:image/png;base64,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)
At t = 10
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
So after 10 years the total amount would be Rs.1648
Question 22.In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?
Answer:let y be the number of bacteria at any instant t.
Given that the rate of growth of bacteria is proportional to the number present
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Separating variables,
![](data:image/png;base64,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)
Integrating both sides,
![](data:image/png;base64,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)
⇒ log y = kt + c -i)
Let y’ be the number of bacteria at t = 0.
⇒ log y’ = c
Substituting the value of c in ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAwAAAAXCAMAAAAIul6NAAAAAXNSR0IArs4c6QAAAD9QTFRFAAAAAAAAADqQAGa2OgAAOgA6Ojo6OmaQOpDbZgAAZrb/kDoAkNv/tmYAttv/25A62////7Zm/9uQ//+2///bPqvS2gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYElEQVQoU52QSxKAIAxDo+JfRKv3P6ttAS0LNnbBzJskJQD8nTDuMXo2C0ADHxlA/WbWhvYQl9jYqBI5hXudPoDvLEgo2eDrIBltAFwzbyMXLyoqaIM0VsitRXrfU/uQB+q7A3ZPveLGAAAAAElFTkSuQmCC)
⇒ log y = kt + log y’
⇒ log y- log y’ = kt
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJwAAAAqCAMAAABx9cIXAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpCQOpDbZgAAZjoAZjpmZjqQZrbbZrb/kDoAkDo6gVg6kGYAkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm2////7Zm/7aQ/9uQ/9u2//+2///bVio6dQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACTElEQVRYR+1XaVPCMBBNQKgHYItCVYpaS43t//+B7m6gR6C5RqDOdD90ZEx2315vs4wNcskIlDHnk50I+GhzSbOWtsp4smMs7yU2xtLxJ3ymlr5c+JgI1qx86WNSIRBlPGU5praXAnlN1r1EBqCK6PEJyq6nkvCwp8gAVo792lfJr8kjqYb+gUSseUSnpyPw6fzDlBIN/5fx4s06cNZzJOfQ/vQR9yYi0ILjc2uOq/SIWz1p1+CY6eifTc4KnHUIIadpm+DFg5JoqUssOZ/hf8ptwEHcpwLqSTm/ecXrmhbPOWSUPmhWibLqGP0WwYr9xKgyGW/Yl7zpJqgHpt3OmAqc12CQTJQxsmgRoUNS2i8gApdg3dOUh2EKh7VtkJzUhHrymcULqwlOGm5KOmoGBpUWkXQgPIDDn7U39JcpjqAnQ2xq5I5caYNT66cLHEUtAayZV1oPlWpqiGNw+rTWkYPQQd5X2jB1plUEmCR3cLW1Uw0hn+PkUooZ9RGqOeJXw6teU3PHVCIJO4RuBYSyfXyEgGFRiCBk7xoebpDwvtg7zQHjoKcZsNsCyxmoCoT+dBLSgzURsi1vdZyiBpY5vqYPEoRhmrTvFku88y1ZqJIicidlK9eUCWG4Q+vWUbT3lGllz+WQW+Co+GCEKePnTC9O86tE8TS7g5LDh1Zz3T/P5mrxnutMQ73ul7HHrHVJr8fZat0vnp2b18Oc25Vh3XeLV+P0sO57h25Y931Dd9V1XwfaZd33dd77ntO6723F8yLMVvt139PGP7j2C7xRMiLnQU5DAAAAAElFTkSuQmCC)
Also, given that number of bacteria increases by 10% in 2 hours.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Substituting this value in ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAXCAMAAADa6lTVAAAAAXNSR0IArs4c6QAAAD9QTFRFAAAAAAAAADqQAGa2OgAAOgA6Ojo6OmaQOpDbZgAAZrb/kDoAkNv/tmYAttv/25A62////7Zm/9uQ//+2///bPqvS2gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAaklEQVQoU61Q2xaAIAhbZffMov7/W0MsMn2oh3jY0Z0xBsDv5do5eK7FcAI1/HowoHpKJ7ty8XruEuAKIjL8EQD2sUsY2CpjvFHcBfuF8T53ZmDreRYZziKgeeLckjmubK9rdxXpfd4vfwC6+QTJn7ZM0AAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
So,
becomes
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, let the time when number of bacteria increase from 100000 to 200000 be t’.
![](data:image/png;base64,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)
So from ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABYAAAAXCAMAAAA4Nk+sAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6Ojo6OmaQOpDbZgAAZgA6ZjqQZpCQZrb/kDoAkNv/tmYAttv/tv//25A62////7Zm/9uQ//+2///b+8IU0gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAlUlEQVQoU71Q3RqCIAzdysLUEiWR6fu/Z9sgkj66javtfIfzM4C/Pd89k1dAe3ClW9pKGKidq9l8szIeEO0+IV5WMnhe4E0nY2GfhBIE5bkXFYHBn1jR6w7ummEyPWz3aOZUXNmiEpRVwKLrUuIDG7ZhfMjfrJ3qOIyGTJCB02r7mE6Aes3Y8vvVyfmCBf1z74rQT+gFEfYIIKowb8YAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
So bacteria increases from 100000 to 200000 in
hours.
Question 23.The general solution of the differential equation
is
A. ex + e–y = C
B. ex + ey = C
C. e–x + ey = C
D. e–x + e–y = C
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
separating variables
![](data:image/png;base64,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)
Integrating both sides
![](data:image/png;base64,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)
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Or,
(c is a constant)
So the correct option is A.
For each of the differential equations in question, find the general solution:
Answer:
Question 2.
For each of the differential equations in question, find the general solution:
Answer:
⇒
Question 3.
For each of the differential equations in question, find the general solution:
Answer:
dy = (1 - y) dx
Separating variables
⇒ -log(1-y) = x + logc
⇒ -log(1-y) - logc = x
⇒ log (1-y)c = -x
Or
⟹
Question 4.
For each of the differential equations in question, find the general solution:
sec2x tan y dx + sec2y tan x dy
Answer:
Dividing both sides by (tanx)(tany)
Integrating both sides,
⇒ lettan x = t &tany = u
⇒ log t = -log u + log c
Or,
⇒ log(tanx) = -log(tany) + log c
Or
⇒ (tan x) (tan y) = c
Question 5.
For each of the differential equations in question, find the general solution:
(ex + e–x)dy – (ex – e–x)dx = 0
Answer:
Integrating both sides,
Let ( = t
Then,
So,
⇒ y = log t
Or,
+ c
Question 6.
For each of the differential equations in question, find the general solution:
Answer:
Separating variables,
Integrating both sides,
Question 7.
For each of the differential equations in question, find the general solution:
y log y dx – x dy = 0
Answer:
y log y dx – x dy = 0
⇒ (y log y) dx = xdy
Separating variables,
Integrating both sides,
⇒ let logy = t
⇒ log x + log c = log t
Or,
⇒ log x + log c = log (log y)
⇒ log cx = log y
Or,
⇒ log y = cx
Question 8.
For each of the differential equations in question, find the general solution:
Answer:
Separating variables,
Or,
Integrating both sides,
Letbe a constant,
Or,
0r,
Question 9.
For each of the differential equations in question, find the general solution:
Answer:
Separating variables,
Integrating both sides,
Now to integrate we have to multiply it by 1
because,
∵ { - i)
So,
According to i)
Let u be and v be 1
We can take the values of u and v from the formula (I.L.A.T.E)
So,
Or
putting the value of t,
Question 10.
For each of the differential equations in question, find the general solution:
extan y dx + 1(1 – ex)sec2y dy = 0
Answer:
Separating the variables,
Now Integrating both sides,
Let &
Then,
Then,
Or,
⇒ log t = log u + log c
Substituting the values of t and u on above equation.
Or,
Question 11.
For each of the differential equations in question, find a particular solution satisfying the given condition:
(x3 + x2 + x + 1) dy/dx = 2x2 + x, y = 1 when x = 0
Answer:
(
Separating variables,
Integrating both sides,
Integrating it partially,
Now comparing the coefficients of
⇒ A + B = 2
⇒ B + C = 1
⇒ A + C = 0
Solving them we will get the values of A,B,C
Putting the values of A,B,C in i)
So,
Then, 2xdx = dt
or,
Or,
Now, we are given that y = 1 when x = 0
So,
C = 1
Putting the value of c in ii)
Question 12.
For each of the differential equations in question, find a particular solution satisfying the given condition:
x(x2 – 1) dy/dx = 1; y = 0 when x = 0
Answer:
Separating variables,
Or,
Integrating both sides,
Now let,
Or
Now comparing the values of A,B,C
A + B + C = 0
B-C = 0
A = -1
Solving these we will get that
Now putting the values of A,B,C in ii)
Now integrating it,
Now we are given that
Or,
Now putting the value of
Then,
Question 13.
For each of the differential equations in question, find a particular solution satisfying the given condition:
cos dy/dx = a; y = 2 when x = 0
Answer:
Separating variables,
Now y = 2 when x = 0
⇒ 2 = 0 + c
⇒ c = 2
Putting the value of ini)
Question 14.
For each of the differential equations in question, find a particular solution satisfying the given condition:
dy/dx = y tan x; y = 1 when x = 0
Answer:
Separating variables,
Integrating both sides,
⇒ log y = -log (cos x) + log c
Or,
⇒ log y = log (sec x) + log c
⇒ log y = log c (sec x)
⇒ y = c (sec x) -i)
Now we are given that y = 1 when x = 0
⇒ 1 = c (sec 0)
⇒ 1 = c × 1
⇒ c = 1
Putting the value of c in i)
⇒ y = sec x
Question 15.
Find the equation of a curve passing through the point (0, 0) and whose differential equation is y′ = ex sin x
Answer:
To find the equation of a curve that passes through point(0,0) and has differential equation
So, we need to find the general solution of the given differential equation and the put the given point in to find the value of constant.
Separating variables,
Integrating both sides,
{Using the formula, )
Now let
integrating for
.
Or,
Or,
Now we are given that the curve passes through point(0,0)
Putting the value of C in ii)
So,
So the required equations become,
Question 16.
For the differential equation find the solution curve passing through the point (1, –1).
Answer:
For this question, we need to find the particular solution at point(1,-1) for the given differential equation.
Given differential equation is
Separating variables,
Or,
Integrating both sides,
Now separating like terms on each side,
Or,
Now we are given that, the curve passes through (1, -1)
⇒ -2-c = log (1)
⇒ c = -2 + 0 (∵ log(1) = 0)
So c = -2
Putting the value of c in
Question 17.
Find the equation of a curve passing through the point (0, –2) given that at any point (x, y) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.
Answer:
We know that slope of a tangent is =
So we are given that the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.
Now separating variables,
⇒ ydy = xdx
Integrating both sides,
Now the curve passes through (0, -2).
∴ 4-0 = 2c
⇒ c = 2
Putting the value of c in i)
Question 18.
At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (– 4, –3). Find the equation of the curve given that it passes through (–2, 1).
Answer:
We know that (x,y) is the point of contact of curve and its tangent.
slope(m1)for line joining (x,y) and (-4,-3) is
Also we know that slope of tangent of a curve is .
Now, according to the question,
(m2) = 2(m1)
Separating variables,
Integrating both sides,
⇒ log(y + 3) = 2log(x + 4) + log c
Now, this equation passes thorough the point (-2,1).
⇒ 4 = 4c
⇒ c = 1
Substitute the value of c in iii)
Question 19.
The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.
Answer:
Let the rate of change of the volume of the balloon be k. (k is a constant)
Or,
Integrating both sides,
Now, given that
At t = 0, r = 3:
⇒ 4π × 33 = 3(k×0 + c)
⇒ 108π = 3c
⇒ c = 36π
At t = 3, r = 6:
⇒ k = 84π
Substituting the values of k and c in i)
So the radius of balloon after t seconds is
Question 20.
In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 double itself in 10 years (loge2 = 0.6931).
Answer:
let t be time
p be principal
r be rate of interest
according the information principal increases at the rate of r% per year.
Separating variables,
Integrating both sides,
Given that t = 0, p = 100.
Now, if t = 10, then p = 2×100 = 200
So,
So r is 6.93%.
Question 21.
In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years (e0.5 = 1.648).
Answer:
Let p and t be principal and time respectively.
Given that principal increases continuously at rate of 5% per year.
Separating variables,
Integrating both sides,
When t = 0, p = 1000
At t = 10
So after 10 years the total amount would be Rs.1648
Question 22.
In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?
Answer:
let y be the number of bacteria at any instant t.
Given that the rate of growth of bacteria is proportional to the number present
Separating variables,
Integrating both sides,
⇒ log y = kt + c -i)
Let y’ be the number of bacteria at t = 0.
⇒ log y’ = c
Substituting the value of c in
⇒ log y = kt + log y’
⇒ log y- log y’ = kt
Also, given that number of bacteria increases by 10% in 2 hours.
Substituting this value in
So, becomes
Now, let the time when number of bacteria increase from 100000 to 200000 be t’.
So from
So bacteria increases from 100000 to 200000 in hours.
Question 23.
The general solution of the differential equation is
A. ex + e–y = C
B. ex + ey = C
C. e–x + ey = C
D. e–x + e–y = C
Answer:
separating variables
Integrating both sides
Or,
(c is a constant)
So the correct option is A.
Exercise 9.5
Question 1.In each of the question, show that the given differential equation is homogeneous and solve each of them.
(x2 + xy)dy = (x2 + y2)dx
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x = kx and y = ky
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMsAAAAxCAMAAABDLK1+AAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADpYADpmADqQAGa2OgAAOgA6ISAAOjo6OjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjEAZjpmZmYAZrbbZrb/kDoAkDo6gVgAkGYAkGY6kLbbkNv/tmYAtmY6tpBmttv/tv//25A627Zm29u22////7Zm/9uQ//+2///bEpEQfwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADyUlEQVRoQ+1YW3vTMAx1uIwCYwMKDAjXAktpmvz/n4fkG4llWU5tUh6al/Wb4+NzJMs+kVKX5xIBEoHfb5vmtjwulWCKiAxvvqn9g+9FGDC5EoxEY/x5fd81Id3u5pefeHyaoaUSjMQ2OT62rw5K9ST0x+cf3bzutbxCJRh5ocQbY6uJUi3KZ2MflsvYepkO+RSYIt6xyf3De0aL6q4gYUp1pPIjWk6Bqa1lbJ9oSMxL1zSPPzfN1QF+YOBNYnqQ0s/zQLUEMI/eIcxxo6uQhamt5bgxNFELEDqonf6h/6n3zbBt4JG0hDBjizk1G5eFqa3F1Qn+7a9hfWThan1nckYemhcC0+HW7cx8DuYfatmjFNgSmxeuQHamYGbPDtOkn+nJN9FiYDBR4ydzlMdgautAPB0/3GNAzjLvPEmOBM0LgcEC6i3gWlomAT1u9JYY7t5bfWxAk3vMwoC6na2yM2iB3MDisDGGra2TE+tFnxTD9uWdSflq9eIPICQAZ9i4g7Orb/T9OWyZ+57mJYRBBQYkAVO7ZuzFAOUCOtqmeQZ/8RhGHqwNY+8XD4MBcTs1y81V0eWXpGj23s9ahsL07kAPYMavOlJxK2rXSltdlpA1UpHxRfGcw0DRuQM5zC5uBLi/OCtqeSStLh9cg0qfiU/OycwMZmxvv9i0hDCmsngratZKWd0UG8x2ZINNvl9ytOhN416EyrsxAZp+BhmW+pRLWFFOi7O6WWRWeanfoLtLOVqvBRxuxKOuwjJzEZ2XlKP1WtDqEo+auco6r2ktSUfr6kVbXd6jxo3hOiLsKqEW6mitlsUe1ZvelX7Y2k86WnNCWKt7Do+andswL9TR2rxIHvU/3GPU0fp6OZdHXZYX1tH6w9paXYBd36PmasFCgC9O0zGhjjYcWOxRhY6j1AmmRtAKYwcmJjqIwbAlnTfOo0ajJ3Qc5U5wxDyZhdgB58cInx8fZv9KeFRGiti4JF8y8w+YE7RYnyztzIRHjU6VO45w+wYfmOVaZlaUlZTwqLE5cuNSKaGhHOt5+j0WOkQpFSXjYuNSbijHep5eS+gQS7hKc+XGpdhQjvY8/Y0XOkSJUME46TiGjcuwoUy9A0AscIgFXKWpFRqX0Z6nvQtNM3TqECVCBeMVGpcAscAhFnCVps7q5bTGpa6XcKqvl9AhSoQKxis0LrWOac/TNXHdANDzDrGAqji1vHFJHaLVEu1iioRKXqjUuBQoeIdYQlWcW6lxmVhn6hBFOmUvVGpc8iQmDrGMacbsSo1LdqW/DjGDzOWVc0TgDz/+m2mHi89NAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
(x2 + xy)dy = (x2 + y2)dx
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG8AAAAwCAMAAAAow9RwAAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OmY6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZrbbZrb/kDoAkDo6kGYAkLbbkNv/tmYAtmY6tv//25A627Zm29uQ29u22////7Zm/9uQ/9u2//+2///bGBFdlgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACSElEQVRYR+2X4VKDMBCEk9pirVatLbaiVBsk7/+IJkDCwSUXQmHGGeVHp2OX3eRyiV8Y+3/cFfh64nwzvDiRcmRcPp7YefE2NLArF5wfhr4JdMXt4Dz9FpCLUXn5Q9QggbxIRszvjJZPpoQNlI/Jy3G3UHlWXqhGWycHmXK+vBQJp5og05q8Wm2h4kRvOr08aGnlRbJn8qgcZLq8KAu65zL1c+Vabrl66Dxg2cqzKkW/md98qI8V2QN6UP42QfXEluVWN1m1fvpDvgZavEjW7l2uSt08oELYEuTJdMWEni355ETB0fywJcjTBc1C+6Lcveiyux/cn8hSj6BZP9UD9zuvV52g6l1uvUuM87Blvjix7+e60zIeODFkpgTCq3LsP2Qp3/li85lUgcJfqnp2qdqdurU9w3LkkZaC3gxRp2Uj9lqqlQluhshA0lKmm+PE0yMt1drchfZe7Pymt4wcwR+XO8/g62piT/X2C2HoUMf96brBTv32DPWceoi/0m8UX19zIxjB15EXiG6dx/CuxqzeBYLka5g4Mq9PhoPyGr5WMGHZ2d9kHRG6QPTzjHpv2V1TZ8PXzLIz1dNAhC8QaH5GDdi95WuanZtBtCLHBcLBnw2OW9CG/Oll5+5S14Ddv0C4zyJjaUEb5jGKnW0mIXL0S6O2oA3zSHY2eZTIwbsGxw1oA76m2dmsHwXYeP2s2oK25esAO9d5tKifB9UGtA1fqz1CsvMQwMb329YyxO5T/2OZgd29Q5yB3clyzMDugbzJ2X3gcv8APl84e1YuWd4AAAAASUVORK5CYII=)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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Integrating on both side,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKAAAAAqCAMAAAADOqChAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OmY6OmaQOma2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmYAZpCQZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm27aQ29uQ2//b2////7Zm/9uQ/9u2//+2///b8Cr+9AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACbUlEQVRYR+1YWXuCMBBMaG2pV6u9lB7SSg9B8v9/XhMuExLYDQl9Mk9+6u4Mm93MEELOa4wKfK93Y6TtymkLd1zRi307GYu2I3E2wvVhZbPP5B8JmuGAYiAIsojSq0MWGoptXWkDnDtBwqLJgZBUr7U1PzIOQZIEfJKShT0fLcKZYEzrJTjVKwsXJL/1Me7OBAUlfYrFHqfXHgo41haL/ou9nD0jVZDky6cHPifuC0HwK6RyLYr+V5fpoI7pwBGB4Vro6fXvSSjYW8hn4vIZJjj0jEHAweDuG9eZwV41sxsfh0VKlTbpfkIAzpAn1VtuSAFTJEEITs+DmCIM4SzEHToQnJ4n9iGohGAJQnCjEMxWlE7DLcrj9BGs8hCurpNDUnU1i5wrmIUbwl55DyI8Th9ck4fEfC7qcfdAMC6slxgS1eOYbEYf3CmPeNLaKqkRjXNBfhDjky+FpBS9A3ucLjg1D080nVej6V5BiSDscXrgpDzVXhQUvRJUPY7lFssE8/vHejTKxnZZLBK+sDxgQY/TAyflYS+7fFm5zfJrp5UE7+S4LpUE8jh9cE0eFvOuTiu3xKKBrun0TOyDBvOf0rNBHqcPrs6ziWjAK0hLhvmyLVG2r/pW1dfhoPC2u7B+1YcA1N8tvVN+t0/UFhz0qo/lqMNBkfxuYKbNMOQ3irNp0M2CBqeILkS2+R1BEKO6KDxZdFEB4k8Ygr5uFmTRHUpw3JsFSXSHEjTHwaqLxDO85EKRqC32dbMgiS7Ey2pIEKqLwpNFFxVQDAnuNQ9SXQyeIrqYAH7CGW8WTLGQ6iLw+HEqiS4i4PwX2wr8Aez9RV0I65oAAAAAAElFTkSuQmCC)
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- v - 2log|1 - v| = log|x| + logc
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![](data:image/png;base64,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)
The required solution of the differential equation.
Question 2.In each of the question, show that the given differential equation is homogeneous and solve each of them.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x = kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
v = logx + C
![](data:image/png;base64,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)
y = xlogx + Cx
The required solution of the differential equation.
Question 3.In each of the question, show that the given differential equation is homogeneous and solve each of them.
(x – y)dy – (x + y)dx = 0
Answer:(x - y)dy = (x + y)dx
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x = kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
(x - y)dy – (x + y)dx = 0
![](data:image/png;base64,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)
To make it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
2vdv = dt
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
The required solution of the differential equation.
Question 4.In each of the question, show that the given differential equation is homogeneous and solve each of them.
(x2 – y2)dx + 2xy dy = 0
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x = kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMcAAAAtCAMAAAAtBI8QAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OmY6Oma2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZmYAZpCQZrbbZrb/kDoAkDo6kGYAgWYAkLbbkNv/tmYAtmY6tv//25A627Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///bJYHYOwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC0klEQVRoQ+1Za3fTMAy1C2Nha8Z4dAE2Q8Zosvj//0D8SGPPUhrZFVA48ScOR1fSlR+9uRNiXesEkgn8fCflNW0qMDQDTCtRHDXc3IvHzQMFD0MzwFiBTso7SmFqTP+WxMOmg6EZYNBPx8uj3VIZCxiaAQZV+opzPx6J18O0AUMzwHBYrDxaOg0YmgFOaPTmfbmq7nQj5Zt9X8lX38mHIgQqKS/2rbtmnaHRzW9vXOcQGtAL4COd9dVO6C+mAd1c7E0HRTSEUOaJ0o1pf6ilWUeOaagTQg/oRfA8EeW6t4Vb+1qW3jLbHRGL1MlAzzAZavu2uPvRV1sx3JBfzPR4VlfgXpjzMq74BwWr0yPorOMd8bBD6S6z0HGwGzNloXXI6OX9sHdDFb+/w/uP1KuF1MlA40R0Y3fA/w4O9acP5rIULf35YaiJmwnr5KBn2ms39+L51j8wStJ/iF+m08ogOyo8rZOHntmQb3Jz/VQ5IqWPrnlx5cbshyQSSepkoovOywpaJzA3AacX/oP17/HANWvKIxKnZ7pLuGYF+zGJ0zOlMaNZAQ+qvMR13p8gD7TknOQ8iNNJkYZ/LPaJ5ETSnPBfwn0MAc2K3PNT5eUi11MDMM0KeRDl5d87V6hmhfcjQ5yeOthSPKJZUx4c8rK0PTIO0awJDygvuf1EcrPcgcx+om1P/zB+0mvipy4XH1Yfzjel5E4819RPXSYiv4OH/chteX3jo2xHP9EMcLIEmabjv/uTvMFP/MpacPITgyXIRkMoe64SNRc0IKvMC34iVXXRafrjmuad/ETOgpEPZwxFaAnSm4aRuvH2S5I3aEDGgjEP7/CyLW+N2fUyb6QB+QrGPIiqi0pUXY6mXpp30oCMBSM/kcHUiym2zsg3JwvkPfiJrAUnP5FZdbk/V9qcU97gm3oNyFtQj37iLs8SXDxco+bfBjUXeDgNuLqIizM8l4Bfa2hIDz9r5JsAAAAASUVORK5CYII=)
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![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Put 1 + v2 = t
2vdv = dt
![](data:image/png;base64,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)
![](data:image/png;base64,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)
log(t)
∴ log(1 + v2) = -logx + logC (∴ From (i) eq.)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The required solution of the differential equation.
Question 5.In each of the question, show that the given differential equation is homogeneous and solve each of them.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x = kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides, we get
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMsAAAAwCAMAAACIcH7bAAAAAXNSR0IArs4c6QAAAK5QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmYAZpCQZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29uQ29u229v/2//b2////7Zm/9uQ/9u2/9vb//+2///bFzPogwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADcUlEQVRoQ+1YfV/TMBBupuKqwiZD8G0UBYoQh+Da2Xz/L+blpellpU3Slwz9LX8N2l6e5y5391yiaL/+Mw/cn13vgNEYu25OyYu74FxG2TWf3dLwXEbbdQdc4ByMs+s4Vm3Hdpxdx7G652LzgO35Pi42D9mej+PBcaz25/IzJuc2K9vP6SRE399GZt01m/5O/Liwy5gQ8vKrrwN83zeROezKPIn4Aur+vj+y/E2I0wKMMuJ5lG3I6gazICefRyfz5GJFVjMYriLlsV9aWpHVDKbBFK8vFyuyHXHJTwk5is9ZQsjrdR47jT+tXJTBKCXkYE0J4YeYJSHiksfLiH2HfGHJwRoyx2XPVmTaYJRCvquKF4ZLKhjw3Bftjh6jSg2eVcusQq3IKoPcO8qe+YU2O+APgF0sOHhxvPP4OCpOXNrAFpcKkWkQLB7NpWuCxAVx4W7Mpi4NtA0ZMqhCvQsuPFdSozZ3OWOYS/Hxs0o/mYxdFoUC4vgdS3gkZGsrFl8+OX3XhgwZZBfXxUIGWv63wwIb7l/RyVW0OZN9PyU485tttCLTBlkK1jJpkiVulmt7Fm7eVVl5QybzX3K2cCrINmRMGVwmZAJxkf2lWPgJC83JLYHdQ1d3ljcymxjFW7Afh9VtpZnA5Wt0dit/PkCjn/dgAtXbT8AXH+5oLV0aL21Z8r5KWvYNaLF7QPzKyJv8nYhzcXIVrXoo8CeRtboGhNFsq6Y0X9qamZUfwocpWUabhSlIKnd6OtZAWkdWCi/3lGi5tDVzVsiGlAeVbo0ltCzVhlLpddzEx0h4ORtrGBt0ifzzyE3d6JMFbcOQv2U4Vv3SpYYXCa++XNSwsLkkPBhwnkuDXJpj+avOIh2YCpdypfDqy0VNrA+PYqyoKrLkiOWvOHkZUMncz7YTOuuVUs1KwxnT0zdXuhEtYaooYPnL9ThvX753FTY+lfCyvamfV1zgooMvWVr1v7l+1QKmzCIsf8VsMfxCwsvZuC0uXCOKiiyKy1T9QPJ3HC5YeA3HBWKwUiJO1N+M/4Hkr8iXoReUSim8vORjQ4ahSw9K3sp0EeVXOAzJXzlbPIPVfGmLJHhRtno1RgnsupX2afehPID6fl3EIRC674cC1mEfx0nnXwgLH9uQTm5yhtLJHXwV9hNjfnl6az2/hEW23+25euAvHQNpWyANJIsAAAAASUVORK5CYII=)
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The required solution of the differential equation.
Question 6.In each of the question, show that the given differential equation is homogeneous and solve each of them.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x = kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALwAAAAcCAMAAADsiyV3AAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6Ojo6OjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZrbbZrb/kDoAkDo6kDpmkGYAkLbbkNv/tmYAtmY6tv//25A627Zm29uQ29u22////7Zm/9uQ/9u2//+2///bj8FCzwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACXklEQVRYR+1X23bjIAyEtPX2km22l23dy7qlu8U1//+BK4ExwoASfI7P6UN4yEM0aEZiUIgQx3XswLEDK3ZAycVrRVWHpTZPfw4DfkdUf/H5HWUdpkmfHYZbhtJSPtbsrMR3UfJ/t1Ju97NFMJZQ14kXVfjh5p1oHX69io/N3kswg3GEfVPVeVGFTy3f/yDiTVviJjCOsEoMtLEKr67nJom+KYsnsIiwk/L0E8YvFN2DBy+bR9PiV30jc0dKgyNekBS8g9+gzRH4I7I8ER9r8LA5IWQDjXZb3zwI8wJVmPYUJprO2zEEJ3xIwWt3lp/4hFDxdaWdpxo8LEOIMHssnZWMR6BOgEUVxtoUDPgpBYqH1o5rXr0blAGsQbsmNo9sEzRMsBxh31xiB4YdVmAthR/FH0MfJHjYYFPsWcoJ9eBhh0WO4pOSJw0TLE+obItIzLRnQuM5wILBiit00QdpLuFS8GtqRwkcdZ5qcHmzhMP9b7QJjcGZxb8nVNYYpPgxhUVlbPN1h/31g5KCo3LjaZNoyBFCQ4YduBFLHT0Phfy8h3ryawwSvE9R2KCubu11cpeoDI7FJxoyhKYDq2sJH2rzKr7unAk7/KK0xuCEDylKO+wYcGfJgGdzPtGQErZgZ7wT18K8yc32b2PVazRSUYoLevxDSFHagRPGPOMuGOCeLwHPxCcaEsICHfv8W/A2hDta/RxeQGNNWf7TwAbLhwW3Kn0blOG8BmYfnOz2pfjuZoNMVrU53z9Mw/6lNGjLq+IfHjbIiB923C3K2J/RwHV+nVjptbEO2zHruh34DwDVRwl0CFrCAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides, we get
![](data:image/png;base64,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)
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The required solution of the differential equation.
Question 7.In each of the question, show that the given differential equation is homogeneous and solve each of them.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x = kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANwAAABFCAMAAAAW7Xg4AAAAAXNSR0IArs4c6QAAAJ9QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZjoAZjo6ZjpmZmYAZrbbZrb/kDoAkDo6kGYAkGZmkLbbkNv/tmYAtmY6tmZmtpBmtrbbttvbttv/tv/btv//25A625Bm27Zm27aQ27a229uQ2////7Zm/9uQ/9u2//+2///bA1KBGwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAGV0lEQVRoQ+1ba3fbNgyV3Ie2pW22Oe0aJ6vd7lWt2yLL+v+/bSRIgqBEUIBlddk50QfbcaVLvAiAuG5VPV1ogf7ts89yc/Q//443d9cP8gf5O7OQ7eYS4F3zg0LE43f7YVfXLx6OTb3Zt1cKs3DaMZD9zQuFXHnw0/alwvqn7W1VDTu7bLfZV1V7CQEYyNP2R4Vk2VtBRvF1AEu0No5b+3HYLRaAhzwsthwIKr2O34Aljs1tNdzBx07zeG6ZAqTO7jlwlXmclY2/XladM+ti1xUgu9oE7KJLo5z1GFzG3Yf4ccn6JUj8t7MXOCjiCkP4tP3+nY/mhRKUIJd7TrPnopcPdUgkC3NaCXL5nmvl2ZJUjZhHXF2QXVNXFCEVojHrK3x/bDDvd7E4JnE97Eo5YLpWEVKzY/LaKfaMl80UAV8HXHKhri8rN5WgCMnnuq6WJVJXZw62o2rhkeNNXW/8+yvTR/Z35m/o84Iku+t70tVwypEuzUOASPblz6a+dooWIaNyEcvbRxhvvoE6GPuD2Y/N+6o3TRa87559tntquId95dUwK70h26ylRqSewy4NISBKjs3VvgoPFSFJriMdH6gnirfhF7OSK8bGc3ZLBXNBcTUYkA2Pr4hyo+BKjJiEZejSEMIrBxq67ZvPGR6SJuLY8Z2hnFnutQ2VAOje7evH+uo3p0+QpA4XjSwIbX/5XYhdWoCIyoVlypD9FjMx6fjsvnnd3JKwn+5l/w329W6dVDnbZ1Vfvq2fQ8XOBzqfUGKX5iGmyhUhqecilt0vw73ZC+NQzagYovf07ifbq0w9Z575q4EEolWOdmkAoVQuOU9hxwf7BkQZhSqvnEnuUFDBV34PgjSw3dxGzO/iQp3zXRpCTJUrQialIHR8zvqIRIvSVDvnjuFgHulsT9XWxuuf9vBHb7IMZM0b0JhUXIKTiDCqc65LQwhYC14CVBEyrXO+4yPKkcMJs+tgy5jNuTGeq60spgg9/2Bu/tLUtSlv/Z15d6k/eDVBSr8cKee6tABhJhP1Lbz4pWYg07bXd3xEORr2ee3OaJxJxefc6dciXRpjW++dLGRaJzyWM6aLN3I4ycNrGrjQpmPFt8twjfeoS2OUK0FGaIrVbj5U/VvXOcTDCaOcvKnHVIoV3+Jzzw9pl8YoF7JzDjJmG4o1fDTd4D8NaDc35NCcxHHY5Ss+RAZ3DBh1aYxyJUiqXNLxIdZc2Gv2XFQFt8Pi2R5aZwpZPqxKwl533PW6+Ipfchznqcn3PGTZ7pKw1w1lK+inQ8VfPvuCrMBAzkgmCvuu1ozTh7vrB6z45g+xg/gbGUh73lqOflQRIdXff2DF739dvrpFyEG2mzeRcbnMMk8oj9MCk+Pk4xSzKBWeiOMH7v7MrY/7q/+hOy4hcj4sNbTxCqSxORLlmOivTxuvQBqbQ1Oeif7atPEKpLHv4HJMdIFikU6cNb3lCqSx776yTHThwCKdOMubnBVIYxOUPBNdsLto4lw4bE4z2AqkcXBclokuuOfiyq1AGpNxYYaJZjTwE2dK3nClRD5DWYE09oNVK1uGic57DifOFR3l5NWTn8RXII3prpgy0fk9FyfOdO7CKMdMryZmW0wa59bX08Z0bhlHOUxccrt28r2ANNZ3UXramCrHMGBRDGHeiSRogTTWK6enjalyOMrhFoY6I6F4BaQxXaNAREuZaFrEA5yf6YMwOMphjeqGTwKKV0AaJ4uwRDQsJmGik1yHcDhxjuQNoxzSxvMUb5HhzcCzRLQbXAuY6KS5RLgwcX5PyJu8dqjcPMWrJY15IlrMRBPa2JZ74LVVl5+JzlO8Al51tDBLRAuZ6NGx4IwfFIVsSdqfPMWrVq5ERIuY6HRUP5saMz5FTtz9EI+neAWkcQLPEtFxjRkmOuVsPa+tCkt0yBzFKyCN6cI8ER1p5Bkmmio3mxqzSmMkz1G8AtKYLFAgosVMdPILIs9rqxznjsD2muO6As0oJ40lgshpYwna+B448ki4LiSIRaSxVBQZbSxFmyhnOxQJ16UjjaXiyGhjKRqjXJ6WTW8OqVlCGovFKYCKe3p2MflhdQ3SmHCzWtpYYj7NaG8F0hj/x0yGiVbYndFURRuvQBqfTRtLPGd+jCOnjVcgjW2qzjLR/wFtvAJp/EQbi4Lw6aZHbIF/AdahKrtLeb5MAAAAAElFTkSuQmCC)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
Integrating both sides, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
log secv – logv = 2logkx
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAO8AAAAmCAMAAAAbSTz3AAAAAXNSR0IArs4c6QAAAJ9QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAISAAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZjoAZjpmZmYAZpC2ZpDbZrbbZrb/kDoAkDo6gVgAkGYAkGZmkLbbkNv/tmYAtmY6tmZmtrbbttvbttv/tv//25A625Bm27Zm27aQ27a22////7Zm/7aQ/9uQ/9u2//+2///beUZI1gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAD+0lEQVRoQ+1Ya1vTMBROxmCiIJOBCpsoRYUyXO3W///bPJe0TbdkWZLDFx/yYZQ0ec95c66NUm/j7QT8J9DMtT5e1RM9uhc6JnlEIcUYppkfr5SqxOi+BqIY4fX3R1UePSlVnqhqCryzhzxitko9QP0BnLie3KpmAQ/lGRDPHPKImQrZ2zezW3JosC36tCrpN2fII+Zos723OKEZcOgCiQPzy0x4ecRMhezt9TtOyZvZp2v25ApDOWPII2Yos7PVGEOpQhu75hpYHlGQL+YpHr1ZKVcnD3nEZFUcG3tuFccxjF7jFEnyiC61UzujgrIxVCIqRjw2s1DGKveIS0NEuT5U13xia7SZkVWb+fSuM69ptvba1i8uHrF5udJ6jKftQ3XMJ/KtJ2RK6HYvrKpbBAPYLy4esdA3aj1Dka/Pt9JturLNWQb7aD/feETK5yUqEs23Bs84f0SDPUw0DG6V1gutR9gXw+vRV8pIsG6ETOO143NhzVziEhFpmw8V50vw+W/ICB7apfUEPGOOnlEc3atnYzv89mnugDq+fsH6Q+uojXRbkg47HL9OcYmIFELE14GK89DwrlRBD2QoXEqeQZ0/dMOQZjkFUbqtz2E1cKY3nEJxuLVz28g+AL+4EGKBfkfDDhougT5UnK+AAGXSkjIOTnEdwV/D15SVB332u+WNXK1602rXKcHeatnXrZ9fXBjR4TimpfOhwvwS6aIxP067gGIebEGgv+y0Xp7q8RMcBLE6erL4Jkabdbzb4lIQCWNgtCFq1aWiziGHR4MFRo9u+pN8mZz0bYQoXwKzxIX4uvylgNDcddIeFcjVQAB99Poz10oOaYxLCgV2ch4YuujG7SmyB5h3Xftse9lh9dctzt2M7kWkL+4KNPaRoPhFb4Um0GQlzufwibNG0oNPHEp4V0CxgsLePENBKvHhJ1UU7je2Rp/PHC/ZEl5x8Yj0/dgUyNeDSvMQomYRraR0t4SqSxdQUKRg8F3UegGz1D49wwXkF/Mw/oHvLFP31JyTA+Z7xMUjGg+/9JIgaRgy7+EvZqEdG22uML3/dTrr0GQuU7pN5DM1zA/EiSCisMNJ8G3jAd857hYuuhcfiHPtjkZEvoeToECGnjLY9rvPJBy+W7YeiHN9TUYjco44lAQE8imE7wV20qHB15EQQ21nCm4R6iZ3MAfiRBBRxOEkQiSt94Zd35lmX8jKI0bQCS9lc/SdaYJ5t4TII4ZZRKzg+8S2M829nUQsecQIOsGlzYJLNn2vwD/BDcEF8ohBkVEL/kD1Mp3p+lfUTt9ieUQRtXqQvjOVApZHlNIM3Zjb19A9bIREecQI4aGl0Jx6OtPQTt97ecRUTf6Hff8AZu+UINsZLHEAAAAASUVORK5CYII=)
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The required solution of the differential equation.
Question 8.In each of the question, show that the given differential equation is homogeneous and solve each of them.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJYAAAA2CAMAAAASw5A+AAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZjoAZjo6ZjpmZmYAZrbbZrb/kDoAkDo6kGYAkGZmkLbbkNv/tmYAtmY6trbbttvbttv/tv/btv//25A625Bm27Zm27aQ27a229uQ2////7Zm/9uQ/9u2//+2///budKXewAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADOElEQVRoQ+1Y63rbIAyFbK23ZV3WdF2TdIvbXbxb7IT3f7kJgYVsJwRiaufbyo981MXicCSOJAvxPPYysP30lZ6Xs82ZsFS9WaullBebKpOTdTH9fha4dvOFEGp5ASyVk7UQhZ6NP/JLjaF4ASQVeqqW1+ODEtUrYEiIKlsItcJpqRGOPQxZwNGlKI37zoEuzRIO8GLupmOThTGlx27+/tZOCep44HK6d7msQ303HzjoS2n9VPOwm5vQakS6UYsBRwdWlRExJQEU+dh30eIEabDqYMJfy6qhULb5PUwiSxVepnOdUQpteLuScjLb4Cb650cmZ3Zb9Kpazu4dWQwWrG653bMjSxV+XHBqtQSdhGBR9xAweMeqbLoWhdnO8gIHfcfCyf5TL4i6lS5VeGFpOAUED16t6i3BQmwYVM5d3BCjKAoWSxXGHPjLDgoLc9YrdNaDnH6hs+NOVgRqWPQ6d211I+VVtgiNmUaq8NJFbPx8LV+CWFonOlj7Q8eArbI7oe7B2aEx00gVPly724/1Xf+VQUxHwUKlRdyBMdNIFb6rsVqjXuqwEnqXDqz9oYO6ZdxMb3ABObylSxUeIcnBcAk5RbtjewP48Oz4Y0OeySmzgzwxWKy88MdMSFEEoToBtqS83q4yqRUAqmK5wB98jILF5Kre0jxksHh54cfFUsWRA/j/bVO1U150nENs7gQrLzzOaaaKXrCwfNcKY5VXz+2zYvJZbD8Y2Q2KmWaq6AWrLmJq5UUMtkp9gIT1J0NcIYV0K1X0gkWNTq282mPdNJgqZoKxEgjKQ+2OrFVeBFvut9CiIOXtkNUqL/rtFv429j5AialUu31P0pgJh6VWs42qlRf+CH/ziVf+/uaU9/GJ9/ofzR+o7waggipDNzmy6543kj8a4ODptxjPienP8s9bjOmqhyQjpqseEFdU+zocrjOEZbtq3bLbTywJ6TjZKHXVjUI/GTLePcQYdV01L/RjLHjXnmiU94mu0E+Gyn23iTPJYR344BRnsLN6/1esI0Y5LPaJpScU9vppRk3Lj3JKhX46TKcbpa6aCv2kqOruIdaosl31HRX6sRY86913m4RGn021GPgLGeFXv+yeT74AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
Here, putting x= kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIEAAAAqCAMAAACjpchPAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOjoAOjpmOjqQOmY6OmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZmYAZpCQZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttvbttv/tv//25A625Bm27Zm27aQ29uQ2////7Zm/9uQ/9u2//+2///b8I2C0QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACFElEQVRYR+1Y21bCMBDMVkUjIiB4g4JS1Cr2lv//OXfTq7W1SZoefaAPOZxDMpnMbmcCjB2fugKH5W4IUZRh4wWcvNlnoA4bXb/6AzDQgx2CAYqqAasxVadaGrAaU48MdBQ49gGppdFevjOIJ6rCiicOAKePejXunD0QbOe+1ieEAGtboIZYoT0GzAwr4tY0YGZYZquaC6ePFS0AJnwtXIDzIOK9LhEZFvMARoEPoNRcEV8xscU+EO4oYCzsc4cosJiH1iNctdJ6cl/qROks/izjjcfIHnUfK7HoOARVgLR8wM2SOW0paxfxGUtu2m2ziVMJ/B0LwSZTpRpUGRDv8EJtWeOsymkyRVXAqquwB7yidAZVqGIlt/eKHSVcOnXqIsn84Q6bwvipYInNLpkr6uk7zyxepp7oQd6HZiwKLOEhUKiIJvbgTD+5pNDrVcT1OdbKBQc1qPqBmVebCdG86u8Z2DyNHla8AaxyIDObhncOU3RLCwlQOqcMgDb/JY8RW7TdzPPGO+bbSYBCiI4AkBYRXRUMMuutJYCerLXZeQC0gexh/ELfZRogg5RURwLocOoKgI9LOMM/DWoMeidAlWLnZf3A0RxrDMh6ygTQOfHPub8HALUAo9CuM+ibACWRjgCgO0u8QA2kI8mBWsBCAuQUugIg3uDvpGuSAN9YOZBZE4W+CZAxQGtJA6BfovXrg3+++gtcgzqD1XIeowAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Integrating both side, we get
![](data:image/png;base64,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)
log(cosecv – cotv) = -logx + logC
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKMAAAAqCAMAAADoDRuiAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjoAOjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZjoAZjo6ZjpmZmYAZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmtpA6ttvbttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bCoLcQgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACnElEQVRYR+1Ya1fTQBDdqVAiSJVUVJqiDSpG26TZ///nnJlsUtPX3KXVUw/sh4RDb+7czM5r49zL+h89MB8TDa4Xpyy9oHcL/20wO2GNVXLB6vz032v0GdFwUSVk+qcwESsHr1iP4nSfDTnCSlOAz179wA2irCBjIbYL2ce9q07lXeAFsoJ8VTJBoixSI8gKavTZhSttH8XttQNZQY2OtyWf2ODcDNkeB8hq21VEnb69BdKhpJuY2gOyghpdrtat5bPBnYuo4RirZbX9vcSqin9IiM4+otkNsoIiS7PwgEQ92PFYubf9hfa2xjp/f1AD9dno/nA3ltQvDD3W5ZiwYNq1W9xZjzBtbWpcsVbXj9p0TnxBGiuZTnk75H71yG+0nPI/RmiK7nRBoCmJ91ouPxMabQEjGqvkzi3TidO7zC8ylvh7uwEa+9PRSHdm9suZK9YiUxkQjXlQk0t2CF+dctWurnb4Mad2Ge2vowkaVeiWfrCmcRu/UmnTk7teH+jye+OkTo/hsw5HfwBbmpXGYKvPCvixr1EmEud+vabzIyRboNnQ2H/haI2t4rmeT7YteK/lYaU5XGPjOT4v6bFA+DQU2yh9euHqaCyNwFhXEE8sX2dcHm44r1moJvj46f2lTvXZjkZruF42c8Z/5lGEzj5ZruC6df5FopDhUhWXU/7jgP4SNLY0fKykiV7qlKBBzxL88vtz9QAXmuGikIa7f6G4hiUObfpezns+sySyWRAXREahLZFSNQvg0IXiGntxaEsjl7Q32waqjedQXPNgHNoUiX4RQ3GNwTi0IbK+/QDN8iiuMReHNiTyGS60jf1AFBfiEWQ1N1mDO+d8ab6T7F0oLkgEWS2rIf/4Cy7QZfn8COGCxCg0JPSZg34DmwNG6uX/hecAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
The required solution of the differential equation.
Question 9.In each of the question, show that the given differential equation is homogeneous and solve each of them.
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Answer:![](data:image/png;base64,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)
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![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x = kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJgAAAA6CAMAAAB7yGD2AAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZjoAZjo6ZjpmZmYAZrbbZrb/kDoAkDo6kGYAkGZmkLbbkNv/tmYAtmY6trbbttv/tv/btv//25A625Bm27Zm27aQ27a229uQ2////7Zm/7aQ/9uQ/9u2//+2///byNoeiAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADTUlEQVRoQ+1Z2XabMBCVnMZ0S9yQpg5Na9yFuq0poP//uc6MhDTkYCNkOfBgPXBsyxpdZrmaC0JcRogHSikfQ9adf005V2BVMlOPzRJYdS/lTfKoMimv91UiF5uIqXOC1SpZC/UEOaay670QZVRc4gSrOcHB5C+ufsJlGdFfaCrUapPewWrKMbyozzEDicBCrTJgKluKEv0XdYRaZcDQ63l82gi0ijdkckw06YcHSLPII9Rqsfgq6o+a+XOJCRd7BFpVW7lY/U0IWYl1GX1EsFrG5gq6yROtAlHE5wqkn1Otqmz1FN9hEazCqXYbm8PwRDqH1ehlcDE4mQdy2Y7IzZDHHdmt3YeBVT0rzvCTB/Q5/uWFQll/+W7vvlzF56xQ11ZvN0waFO/OcV6HYGtS6B+Y4ChYrzqpEs/pIHTSQGWs95pQiVevSWUwacDbpAkFr3YYxNIKDu6y6YDZnZk0oLCCE7USh74alHjx0s99DAjBBYcGa5W4yOHcUVl8mSSKI48dSG/TcNJAKzenxLFkCw81on6Dj1+NEcaHnzs0qW1OXc4Td3BdWSU3Kw8myuVa1OkY3XIYWJVYTzBpkINxDkwUXk0E1VEx5lHfYWCGp55JA8TBgTUPn7z9gBYHysVNa2BYZe/xVFTbBPsR+zQHyaIjOBAYU+IAm4V8IKbo7aFysdVEwLDK6oyWXW3EL+1xE6Nn0oDCYZW4yiHcpacW1xU9UC52moBRBhDB41MJ44L+5KEAt0p8nUFRN6n0QtaSMy+Xnh6pnUZgOmfwaoBR1rfAbKtJgQo+IXX8ueEDgTf7OmC0EkO806Hsh+BXhD175kvDigPl0k53PYZiUy7WR3wTCow6phJCMVAudhqB6Y6LcpOxeP8hTZU1flCjgqUyUC5umgKGdVUjum730HfUuHNqFDqT5XcQkWPl4qahD8Wy3AF7UUcPnQIM+uiylUHo/XEUxLA/N/eY9//0yxjjnA5Xs3MqbIfAVbrT0fRh3yvw1ib2uwZfnETiaqsT3ODrcHVgivnuf/h/uzeQYrdGS7aiyHE1yabph4VhyYvrtynxGRyWq2fiMCxMPN8sGXdk5ZQOQ0yrvSVj+DItms7uf35Yrq6/zQjXBUoMD/wHyxReWd3zRIAAAAAASUVORK5CYII=)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
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![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides, we get
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZ8AAAAuCAMAAADA3eTCAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADp8ADqQAGaQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAezoAZjo6ZjpmZjqQZmYAZpCQZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29uQ2//b2////7Zm/7aQ/9uQ/9u2//+2///bYs1goQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFB0lEQVR4Xu1aeV/UMBBtQJFVOWQV8YJ6UISCytptv/83M0eTprtNZqbNJvjb7R+iMnnzZibH9KVZtnt2GdhlIEkGfl1cJ/Ab2mtovAQpGXS5PGf7d9HJhPYaGi96QlwOq5PbMn59QnsNjfdkyiOIJKjPBrymiSJGIdNEFtpraLwYmcf5SBNZaK+h8XC5i2GVJrLQXkPjxcg8zkeayEJ7DY2Hy10MqzSRhfYaGi9G5nE+0kQW2mtoPFzuaFb3M3ZFGyH7670Y+sEqt8leVwAn49ETRx2xOPyTU+vTfJ8xxp59ofqi2ve5BfDaAwyARw2Ibt+Qi0P3MXZEcG7BAcdGhh9XvYyxT7V8Foy0lULciHBZBgCS8fBpxlg2P4+Uplme3Br7RZRzRLtbkOoDcqPBZRkESMXDZB1t0+RvHjlFkaHqtTlx4nYw1Yxy1IHcaHCwaEjFQ+ceYdjkZ8JKTZFuoRdRdWhaAkBuNLgsgwCpeIi0o00WvUKUB3wtiQeijMYHDatzxo5nV03O2IvHaoa4QPJyo8P5g23xsoKxg8eSdlCCscMGTX6olo86AfQCavJY66eaXWbNN+69ycXc6E+XQf5ebnS4jHt2B2vwsoKfyPEbPb1025/tbuelDBedYlHIqojZIV8KS7nbeh9vfXxwfAm0T7/58QF2eGL+INhB7Im/162LrlMhl9NKfUxcgf/CHdVzURDpvZqdZfVbq6135HMlnR0nAM6ZmT6gC4/zOz61MdborfOdbrJWH3UARdvfrPqICbpQ02P0+hkB5w3WwoulXfVj172qWT/p6iPOngLRZ/vmjp3PNbgR+5uNV3/4FOtQ7mrkXD+6kYNm88Tfq/5Edff1/PPHtn30oapGYvgZASfWjxPQwmu+XtdzxPKemJCV4c7zx8nEyA0TiFhKRbn3I1teqO6xYHB3INLpyRIdzg9o8JqCc1ugCE5IzNpQva+1EoZaz4KyK1NSbijZRPmnUyqaG7Z3+lvdZSCaay838UsynC/YDu8y5yHXcxa7QO1k5O+FPQGhnjsOAi03jK2P/jIT0jg9c9DJbey8DQ44lsjQuGH9wJm+1hxSFB0MrS8zjVJBDmZCaYd9BQckh+QZ0NvJFNP63V3p2OL13j+uPvaXmSOT4uE2LivBAcfRcI9S+rV82lOBb3YndodkSWPmuJL7m9CmjsSdRHMjLk7Z8/eghma0Z/cJ5w1wlRs/KUGfNMBEQpunQEP3Pz3zThrT60b+FNrUUmpXxf51do/S0Lq7gVapmDzd0Lod1lMaoQ3LbtDOSGO9+sgMixUldz35cgBqaFZ9Qr1hgT6JoacR2ogk++ZGGtPpFXVSvbj4s60P/6etoQ2+rm+gPmu63aRQ5ZTrC21T8TY/3khj9vpR9ZG1EVvCg2jQQQ1tA/UBfZLz8x98XrUSk9ay1usjqyQO6b1LMQbS0MKfP7BPan2SCG1Ukn17LY3Z/Zs6mOX/dPcikIZm6qOVimm85GjIJ9FFGqGNSHLVvJXGzPuP2MyEGrUURbJ7ZUBDM1vHyPefwTBwuh0yA4mENiQ7l5mWGdRPfh0uGuwH/tZzKt6U+L08f+RffRqa/WXmeP1gnSJOt8NlgG/VSYQ2HDvQavC1sj4Xat1fykdSIZcPSHqbDCy5wYStDhXKiWJ9abdNyYsR68D9j7wWaW7wt4v2l6oxOG+7j4dX/Pixvg7e9nzs4t9lYJeBJ5qBf/KaotGisWdRAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
Put, logv – 1 = t
![](data:image/png;base64,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)
![](data:image/png;base64,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)
logt
log(logv - 1)
∴ log(logv - 1) – log(v) = log(x) + log(c) (From (i) eq.)
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
The required solution of the differential equation.
Question 10.In each of the question, show that the given differential equation is homogeneous and solve each of them.
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
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![](data:image/png;base64,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)
Here, putting x = kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0f(x,y)
Therefore, the given differential equation is homogeneous.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
To solve it we make the substitution.
x = vy
Differentiation eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Put ev + v = t
(ev + 1)dv = dt
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
logt
log(ev + v)
∴ log(ev + v) = - logy + logC (∴ From (i) eq.)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Multiply by y on both side, we get
yex/y + x = C
x + yex/y = C
The required solution of the differential equation.
Question 11.For each of the differential equations in question, find the particular solution satisfying the given condition:
(x + y)dy + (x – y) dx = 0; y = 1 when x = 1
Answer:(x + y)dy + (x - y)dx = 0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x= kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
(x + y)dy + (x - y)dx = 0
![](data:image/png;base64,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)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides, we get
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAL4AAAArCAMAAADxAcJfAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OmY6OmaQOma2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmYAZpCQZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29uQ29u22//b2////7Zm/9uQ/9u2//+2///bAAnvqgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACtklEQVRoQ+2Y2VbbMBBApVCK2UKBUsBtwRSzNTLW/38dkmUHbWhGiyEP0ROHSJnrsWZ0I0K2YxMz8HR2uxlYKSCvp3Tn3sDn9fVXPI4LgqDoDv+1G4HvAUHgiykAPq8p/b7qKvsl4b48ZpYNglsLZZ/XuytCmPWOcN8dNWsefNIuRG23x1EoKZPz8Rs6DYk8jq46Jv3J/O0pH18CO51H7h62l5LPuDUz4ct933xCM50Lv1/++imqd+6BwH+oqJ3HoTK14Tm2GjpL4VowNoibLrb33zxT+d9KFOu3yzD+PF3TgPGAOPhf5AP+XRcP0+2Xan+MOpswtjQAGE8EZm3z2IjafJaND8G4ERC1jX2irsptpRCMG6EpJy75+BDMbPjdKaUH1XWmi4bwxwhESMzuqh3rjNdFst9VV4T/EXs/y0VDMOsIpBHVOrWoQvjNINCydF0X9Tqft55CMO8RZIYm2TVXrOUy5g9B0i/lETzsTLSLep7qIxgzgghxcDQ+fpnsa/g5LhqA0SKMb3h4gOL4rosW2Tw6fn9+MRWsKrXcwWvp/upQyXDRAIwWgf++7Zfjbw317+zRLm7I65k6ddNdNASzjsAbUWds9F1eZ3jvi2j2qoj4HV0cPSvzTnfREMwU4aqmC5F9qvj7pX3Q4y+2+pMb8lhOmXww0LawHc+52AIctpywys4bZ7/9j/vW3PruxRaAX+6yxIUBU1/RQ6fvWNZn45tW8zidH1Ao+HNxaWfCGHIDrx9nAPiG1bTl6D18utwUw9eshgl6luv4H4PpcpOC7z8v11Yj21f+D8QAmCY3KfjKKuz05lgNmkJNhO9JnC+E9v5wLn3GDZs4B97lBv3cMH6G1aAphhevyQ16ofXCfH0/3WrQFJJelxvcQvdiy4efbjU4imGWOGA0uYlYuJ26zQCYgTe8bk1/hNx5HwAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
y = 1 when x = 1
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![](data:image/png;base64,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)
The required solution of the differential equation.
Question 12.For each of the differential equations in question, find the particular solution satisfying the given condition:
x2dy + (xy + y2)dx = 0; y = 1 when x = 1
Answer:x2dy + (xy + y2)dx = 0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x = kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
x2dy + (xy + y2)dx = 0
![](data:image/png;base64,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)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ4AAAAuCAMAAADukVA8AAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6Ojo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZmYAZpCQZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tpBmttv/tv//25A627aQ29uQ2////7Zm/9uQ/9u2//+2///btwvrUAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACdElEQVRYR+1X2VbCMBTMRUWrAoLiBnUhLlVamv//OrM0SZMW2iZpz/Ec8sQDmTu5mczcInRc/XXgZ7npD1wgu5fY3cHJd7/0PEpk06+kZ3qeJfqmR6/Gp4TP3paq8Cnhs/dI79g9nw747B1Ce6PeUyNxLUHeIgA4fW7ZBpe/DVDChdbQe1KAdYiaoXBsLmkYeigUjsUvi4J0D4XC+T/0sjuASbQmMcD5Nouch8QCB2GA8TYJpGZ6HytEXqn2SDzeIpS6zogKB2HqhSQOoxaEOSn2NLjFJnN577QLxWplvRqHnZPDKADHHxQiXzAgLuksmqP89kDQWIR1UROHAk1mIXzKhGWnTi8ccUvHLO7BEcjcVoalusNaMt0ut4yT3z+6Ktg+E4lZv4Sd5ounBypEp1XCIS+bfOF6C3bxZPSOdkuRGhjUw+jMUeEQTEHSChL5vG79WZtMv2R98gGj2W/E+Tm7Ct0rcVYxjGj3wOJH4pvGiyE/1ILP2NvMrgL5Ussuk7j+Xgx/xLBCuwWXbXbZ+6haZr7vXkx6TLCJ0FnC7LjfpWNSPJya3KymSzH0+LevMVx1TBaTTDU3q/Sw8KR9aujQ0MZwVTGZFrlo5SZjYb0BNZJhb2tS4brvSComJT0jN+t8XzeNZ7jfagpXFZPyk7aSm1b3hEb5CkBPDDkHloxJ2T3mrKXcrFwuvlAtC0CvMVxlTCp6dm6a3eNukgqHDKC95nAtYlJ/g1i5adDjZsIzUQ56PtrbE64GZGHHWlOWP1upwUdITs/b96jJ1oVr/YE7p/kAqVFi2tVlvZvXURZtJhYNOfTEQiXvNu917MLx7wN14A9jEkfJEax+yAAAAABJRU5ErkJggg==)
Integrating both sides, we get
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMkAAAAuCAMAAAC1Wc0NAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6Ojo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmYAZpCQZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27aQ29uQ2//b2////7Zm/9uQ/9u2//+2///ba5s8yQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC8UlEQVRoQ+1YbVfbIBiFbtNmm3V2073VuE2cZm5NGv7/fxtvIUACgfTB6jnlk8cTuPfyvF2K0HG9+Bt4vLo9oAY49N0lfvVwMCWA6M3qvjqcEmD0AyphuQCJDnlWeppCokOedVSSfgNjO44xeX73+Nxj8qfA13G3Vi2edMY7xCbR6+W/MkYJ/VlgjF9/jxMN8JVFLAKdRskAIJZ6RDKx5m2OjKlxbMZ6BU4QGyLUeXK/3lvJFLEBQqZ+1BQxtRfKuCliAwSSx9/ur2SK2FMoaS4xPiuuaYnx6bYpZr5qQkoUAiIYn2wrjHlsaQkek6bYIPqD1QktT7YI1fMAQsQ0AiKsylWTy6CECP684sUsqy6MamCXqNZUowkR6xH4ZSkAe4PGmfkH59yuOXWRxU1xgdqP011+TJ+PmI3AIM7O5U3Bx8RQwm+sXqaOxEliBoKKe3YlvEaI1Y1BsstU0n7+qupQViXkoiWPgpxb7frbl5nHB4gZCPTmtl3LqMv/gq5q8QvtruSMJ9is9xSYEDGNQAk7vpYYtPRC0d/vo3/eqlb3mia9w4vzv/KpMLMFh4mhDmFT4gWLiZwn7drnKmj5YTIx6CMbgm94c2re7etOnID5iXki63WcvmBZZpvgDdqtRclBe+q089pPD5WvTHx5YSvhuytZEhVg6wgRGw0J80QrM4MMpyQLbsQ6DR9Ayl6nXWK4+F1i2mBFp3DvlJTTHFqnoRJl9QK9I6Vneb41DFbcadopde+cgXUaKNHumoD3c4OzYbDilGin1CmxrNO4P+rauPB12VZvsOIgtFPqXmwD6+TExBhgeZVIY52wOqekX9GudXKUkKUORF4lvcGKVNM5Ja3EtU62EtF6a5lfeeukN1iRSjqn1L+SHetkKRGdV5if7l0SC5P4nWmwYreqidjnvzMinRkvnmRCCeQ8ccmyuSYN1hw/muz9IGd87LVHfZc66HKGJIqw/6MYL9zvBvfCe7K3ts99n0ByOJ71Ym/gP05AW8pgMUv6AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
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y = 1 when x = 1
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3x2y = y + 2x
y + 2x = 3x2y
The required solution of the differential equation.
Question 13.For each of the differential equations in question, find the particular solution satisfying the given condition:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x = kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMQAAAA2CAMAAACvIsZLAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZrbbZrb/kDoAkDo6kGYAkGZmkLbbkNv/tmYAtmY6trbbttvbttv/tv/btv//25A625Bm27Zm27aQ27a229uQ29u22////7Zm/9uQ/9u2//+2///bbmKWhQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAD10lEQVRoQ+1Za3faMAy12Wi2sY6NdV15rKTdI2MPEpL//98my46tBAcUAyHj4A85AWxZV7KkayHEdTgL5BMpB8sQi2y+fLPL0vGaKyKWUn7gTmbOyyc3zJm1admbZTGTcrjOIrBCMvrBFpNP+gIin0yFKGZD8ECqPJmoN97oD4gYHZi8AAck6rWYsc3bGxDZK4yjLJqKYo6vqcLDGr0BoR0B9r8RqT5IfFecFcTfOynH2tbKAzjgPMXuleUIcU4Q+cdHsTLZGGNBjXzy/t68WmD7oJwTBHpAh4KIbS6KbdJn68aeuM8a9vdddSKV5qTY2YlOQWSRi2edbxnjzCBWNiRsPk1drYyZ+albEHWrJgaDMC6C5Gryqw5yJn2pgUjllsMZ7qxM0ScDCM1wnZTSNnPgU+M1SlePnxFmpRQeKR6wEsRsvCCkZRcIwlC2stP2qQ0DIWKwYjHTIaAOd7GAA475JotGS5EAHEUVDUyjLmj2joSBmtQ4CEOpHyd2WmsUbmJU7WGiVlsqe2tBIJIKqfDbfLdFHUM5GQhQ87Y88eJJjr5iQtWegEdt3xKEcg0OtJEFodi2HiRKHEOhwjIooLfRlJ62tmeJZktq3dVr+RJKWCMIv813B7ZjKAREFj2IYgHHkJy2A0Dk959phvwVQcQeFYRjKAQEFk00iTtt4SAgVdoKpsIBa3IjCH8o7qkTlqE4EPrNbkPzdSsoWvUiBmmpYRDKx5s7+BpNhI9aYNc+mg0dF/FrUDIULwjCh1vpj5MRBMTVADxhLr+beSRV7oR7p5ziw/5k5Ksdt4b3SzqrZCheEJQPt0YRdsc2Rq/USL97iEIlQ3EgNG6dJggf7ggEXqth0BpZfudVgTIUEtjJ4FFsPuki6fhwRyBKPWiN3BkSBWEoBETxBPzmT4Qo+PfbLZBhx8k2N1yNxPZH46AMpYHFEj7c1hWBIKzKtsLxezYeEDU+3BGI0hW2Ru52REUrH4gqH+4KhMB+h62R/F7HFhFTGtf4cGcgivl4bWskfOBvfIqbXWhDWfz+bmvk5pmN4RQNZfbm14khFvDeSkIEnXiNvTy5l307+qB5xHT31T6FL+D3/+U4XYCpLwnCETqAPTDH4R3AHoA4vAN4BXG4BUwHsNbRPlwuU0K1kc5cVJ9mO4CV23qgsJBltEkQsl6tcR1AelsPlRaw7gjbkg5gpaMdoE3oEtpID5NBQbD/aArbqnEV9/+tRgEURK2jfWRVm1WoNtIDtiUdQNrRDpAUuuQY29oOYKWjHapR+3VH2bbsAD5UOtrtlQlcUW2kBwq5LuuJBf4BLMJwoDxtulwAAAAASUVORK5CYII=)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides, we get
![](data:image/png;base64,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)
∫cosec2vdv = - logx – logC
-cot v = - logx – logC
cot v = logx + logC
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAYAAAAfFcSJAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAANSURBVBjTY2BgYAgAAABVAFFWfMpZAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
1 = C
e1 = C
![](data:image/png;base64,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)
The required solution of the differential equation.
Question 14.For each of the differential equations in question, find the particular solution satisfying the given condition:
y = 0 when x = 1
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x= kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIUAAAAmCAMAAADdjKhOAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZjoAZjpmZmYAZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGZmkLbbkNv/tmYAtmY6tmZmtrbbttvbttv/tv//25A625Bm27Zm27aQ27a22////7Zm/9uQ/9u2//+2///bo3YZLgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACfElEQVRYR+1X2WKbMBBcEdvkap04burgtCGXlbjm0P//XHdXCOFWCAziLftgMEij0eyhBeDL/AqoRIj5IY9F9BReqv7gKpkfALIpSAD0B5dnOwC5CK9E8esdDHi2xK36LI8fQG3D+yO/RMwaXF7hXj2mkgVk5JWwVq4eENCCy44lULWUZoS1VPu4BlfJrXeBcnWz9ss1gF9+rn1swTOKP4+lwk9zAAmopACowbvE6GI5gAVFpTYLzvnSbln4NLUrWnDL7D8qmKNTpGnKGXEMXq5aHa+S5WN4KcoVYx6D60LqNCz134PXCixWvO1/wNOOLBkQf94pmXAVIBngsMrvhIgQnK7X70ii2OIDOh/wSfSTWJkhMBmLPN5AgVWZrwlqS25Wj+hqerKnzDRDsGI6dy1ZoVQYGyCNjnoEobijpOOQz68PQG/ovOAbbW4WboV4Qs2sdrSTrEkzfeXfZ3H1Zu6JQSMTDYtjcA+LnoF5zIK3Dp8XYrZDRrzW2a7BwhcXYzzi0AJ57OOF1oWsJ4ue226pK7q26dJDcUEhQY7QuvCruvy5i3WAeiHFBtTLEybhLeZIlRrFHS6c0ZsPTFUzhFi6irWN3uF6fMRi9puiIRaCqkSxxRuuu/gmuq9ueEhTFrtgQ6vhLE6a6dq2W6CTYE8c7PqymOhrw8PMdYr7wgLTd36Qwnn8nChAc7huuZvguilvsxTLuUpCN+HVmg1w/6cAVQAZvv3Vi1pwvxSU3N+WI7Rv1ZhrmAHv6sDbTsCxxNSWv031wYZ/OvDK9Y9pWrE/2PlX4MVrBwlslKtmdezuHfP7gqsUI5MOhymsLzi2yRFqISahMSn4GNX+AggFQo2ycr1EAAAAAElFTkSuQmCC)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides, we get
![](data:image/png;base64,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)
- cosv = - logx + C
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK8AAAAmCAMAAACF8ztXAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjoAOjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZjoAZjpmZmYAZpC2ZrbbZrb/kDoAkDo6kGYAnGYAkLbbkNv/tmYAtmY6tmZmtpA6ttvbttv/tv//25A625Bm27Zm27aQ2//b2////7Zm/7aQ/9uQ/9u2//+2///b+4ev3QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACZUlEQVRYR+1Xa1eCQBCdtTJ6WWGlayX2QktC9v//uWYGBGUXPcBSbaf5AB6UO3dn784dAf7j91RASSH6y9gTvenvIbWLiZL9JUDkCl2A8OANL8duVBdZxt4Y1MQRNSBfJY8hIk24EiiIYOwKWeSZ+Jc3KGF3IhDX7pBFphF1CIcicqeZUSdr2szCn/BEJQf3Tcvb0hTfh0L0Lmo2Upwf6r6SK70GXyW1jhmKq6V6/s45oBXf2KNtrdBiTKXHFdL97BV/t5rgg0HNvSi3kJRvjqlmnsDoP9DEFwqxWVG9vuGOysbeCFb+GEcFvEvsXjSUqfu2Jsx8c0wIDqYwJ5YBPi8R1PgqolEVQcYsoC2g+Sbx0SHis4r6BlQmjt3qYr45Js0hCIwXqka4bUEa38QvqlXOx+zYeOnO15k4fUlXl3PbZxj6KohvgZnxpRSxdz4o4IzL3+RbzrzNl3FhcSKO2rpawZcxSQaLVLWaOGvpwVBfxHznA2qKOnrY2DP6a9Ub8Ube3JbUqZ83Wl1FpBVNdZXql6W7VvU+IVR+T/UtMCGXLDYp1vFG6HwjHrLM/SwUI1BPU6DfrIg0H+phU19b84ho83NMJbMjpgL8kNLJQ+erJO5FlV/MPXH0SKrFDklddzXBD419LWMRpf1jjQnYcjEGHxIfJ77YImzwN27Xh3ctPaCxOCAZ0lH7RLW5Efw/O+tvLjBmxapZ17M/Ni/N7ZuVB1s6HgqaTjoNg9t3mq8tuMHt20J2+/6223ebywr6rlHUSgK7ILrb28W3jGZwe8sZrMKZ3N5qArtgOATqbm83xR9D+wLfOjpTF8rUTwAAAABJRU5ErkJggg==)
y = 0 when x = 1
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALEAAAAqCAMAAADLVkp3AAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjoAOjpmOjqQOmaQOma2OpC2OpDbZgAAZjoAZjo6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmtpA6ttvbttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///blM4GrAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACP0lEQVRYR+1Ya1eCQBDdLS1KS60UM7EyfBA4///nNTML+GDBBYzjnsN8AI+6w927d2bvIkQbV8cAzKTsLa8OVj4gcO+2kXv7Yw/k4GYuRCAn9iD2iN7d8N4axCQKQkxXO0IhVlc7wlbENulYxJU3sEMShNKnxmZVd6M2YVPhIcnRSMq+NZ3CHvFeAqkvaX+3KtiR6GPzYjKZDerxpklB5iLGysh6QXBPzZYvn7fw1eRC5SEO+0vfAHHokPuCqXY1QqIfp0h3Nu/RFL94qtkgFOI0JywcicEGxQSxX8Bu6IxFNJwIvpN5p/4L73XNDyNOc+LmORer2GcbIIaiU4QXY/NoGUJnguYSN+Wwl8exR1xxFIqMEac5wcXsiT8xQHxocE8fyPgw1J2vC/n4rao8BSfP9anMPAjxPmeMWD3qBLGOgiJLfoyYM4v1g+zWPdrtEXNODyewvowqNBwj5g2XqjZKqOJg3cBFFY1VQgNV8AxzQrGaHDRIxyzhRN3n1JD7O3GsPBTlFP6BYTVBHEgaoO9uvhwL+Jyjpxxgr8BHcIGP6h5G2aKmOcE9RJzlL7ODgIsrkreDrBzZ/SD1YsekLhxN8UPd7TFQrSTJiUabAtPDjBpz5+14cbJ7HjfwzmuVXcHMBZxRz25Eu/Av6eO/Q+sCyj9USTep8vLjzUfoXYD5+OSfXEiwaOZ1mKayyyOmFo/F0cwrx8sgrjDJykNaxJWpMx7YcmxMVeU/Wshxk6fDyrymA/UuoH7eNsM1MfAHDU42/uFsaIQAAAAASUVORK5CYII=)
- 1 = C
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The required solution of the differential equation.
Question 15.For each of the differential equations in question, find the particular solution satisfying the given condition:
![](data:image/png;base64,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)
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKwAAAAtCAMAAAAEA3CXAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6Ojo6OjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjpmZmYAZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmtpBmttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bZmEyNgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADGElEQVRYR+1Yi3LbIBAUbhM1bdJaSdq66iO4TZQXqvj/r+vdgQAhkI0tMZNpmMnEyZzEsuzdra8oXtf/zMDTFWMXL4SA7vJX8bC6eSFoAWb7LgtY+QiX+PbYrZp1FmI52xR/qzd3OzaT9beJiIdMkuWnAKJhU1AQ5STYJhNWRZcAsJyxk+eGsU2NH9qSDXLGByudKAFYxa7TzicTjjLggA4xyfrkGXYf5veIWRvVVQxWNrBtiVvh9pQoDUJvUB52jWUQipqPvuibZK1yuS0/kPgQu/xuCwQIRC+XbT8qA1BitCexUWDwHwKlMMlsKGp5wPxU4+quP6sSBjfMPQ0GqkEganGwDaUTCAFuvquI5K76eO0V3gDYQNTSYKlRSr6mH6hhJF+ufk3KIBS1NFidPWsonCtglhFMMepooaYwjloabPD9Yli3Ihj2izIPy5/EhPzzPtrbm/PbtPOCdN26FXk4FIVdLX5KrB5QxmX9aVhnBhu0Z2nNRdYXP3YTG4zS+elub71j33n8bBiykeg1gZ/zibPrdwejbMU2CGzXRv8Rygbv6qg45VgO2PaKrb4WxRZbH2WrKNFA6IjeNKk7d01RLhvv9sK23BSP6Dg8ZpUWrGlSFLrWqe/9i5NrmeVwmfSXB7b/05gmhck1RWSw6UBBSzLbIQzYrsI0QsQRsMY0qb1dU4RPRZZBf/wHRwbK4qJYPbBEoSLTdXCuKZoAOxurSnt4heQxkFlcMWaNadLQrXUyYLPIoLu8seKNadaYJgXWMUVGs7PyOH6Zagp41QK+M8t7qF1tuS5+k0umOqurgTFN5lTGOtlLWRYs1su+rt7DF84vmClbtqJqJeh7mhaKMU0GrDFFiS0sfqIZhlqeU+sq7QWMKTqogwUmNXMMtbyav92ojDTW6TBiI5Oaw17mXFbAdTmmKNV16RdHJjXHD7XGftaaomQ/60rWndSQuHINtQ6pBINJDValrEOtNMTepKbIPNRKAutPajIPtRKxepOapKczB48nNZkBJGwXmNQkPJ03NDipyQth/92Ck5r9H3+NnJGBf8uhVaBJrbguAAAAAElFTkSuQmCC)
Here, putting x = kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Integrating both sides, we get
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJMAAAAqCAMAAACA/pmaAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OmY6OmaQOma2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmYAZpCQZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29uQ29u22//b2////7Zm/9uQ/9u2//+2///bJqPX2QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAChklEQVRYR+1WWXvTMBDUBigxPdyDKzXQqtS0QOxY///PIVmKvLZ1reUnWj/1a7Qzo71GjL1+a2Xg+eZ+LagBJwv0cA1vHtfWlAfanv2sV9eUD7q+Jpn2TNDMcHfdM0Ezw1815UxjZvIzw/+n2v0q4NZbiXqzwh6fMkRBm+3fyqNJ/CgA4O3XnN6RsWOGBFDhE5QpZAinM7QfiMVpIFBq10ViDHPABjWMeL4GeBfT2BA1YQZn8meAeNg57NihjLluW/hHwsUZXSczQI4k8K0yyFgaqJowgzNPQU19RLA0rSzuaXErKoD3+7ZIel0FNRlAxgFO9jWAUiCqaalCEG2xY+K7TKSoTvZSfqzMbgaULAvIuOxrM6EzTcHK8F6JKm6/9uoLBC9var7xmp3fGgdZQHVLgzeNEJXmsQT2D/nPrlQ/9qrb4oJ1l7ERdeRpAB4DSsTTc612oklUqs19H9KkrtWEzlqMUJ4QoEm9QxPfyuKkaFK9xEc7YUntsKbu42fTnrpZj1/dlxd3yUifzqIezK788imkH+UJM4zviwDFt/uu1IkfFat3AcG9mmRn37HDjV5gHPznMHOwHSxgz9poyGNP9ygm/ROuP3InHbvvATbnv/XbJmkRTBmmbSEM4K6CjcyT3k9dGTOK7vKOPWW8ouIMU6Ex09bn0065Z4MY21091knzPFqO3rF0/JDMYGOlYZ25Z2dsaE+mnShqTIJnDEdji7WMgwsbWr1YkusSyNiodxwMrZGSmgXX8jAiY6NqsoamJpT63A2SDcZG1ZRuaFRk/aZY9M0MbRGKaxatsZERkw2NiIyMjRgpj6caGg0ZGxstUp1ONTQSslx82tjSDJyE/ZIO/wOGsUJr2DjTtAAAAABJRU5ErkJggg==)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIUAAAAqCAMAAACqTmg1AAAAAXNSR0IArs4c6QAAAJNQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjpmZjqQZmYAZpCQZpC2ZrbbZrb/kDoAkDo6nGYAkNv/tmYAtmY6tpA6ttv/tv//25A625Bm27Zm27aQ2//b2////7Zm/7aQ/9uQ/9u2//+2///bCDs6CwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB6UlEQVRYR+1XDU/CMBBdQZH6ATLxC6cUdaKbpf3/v867K4HRLsDozBHjJZBs2Xqv796965LkP7YzYD/HQpxOmWlSYpIs0u47LwzVh/y5eOJFQdnLo0ChuCuCVGh5BAWx2YhfFTZDgXKH6hfcEKBLewCiZK6JPgfftIoZhRIUzCj4JdEIQS463DOPbD4OBc7uziC6HZugsJnvxrm4Luxr3EaachGg0BKt0T5HF9VxoYHYyzdccSaxuXov8CtysTmVAxR5PAtOxYRCSzgUZTiCVXeafGBuBff9rOF146m9tBDY6MYGCAUdinAG0/QxKfzZDLjw/MZHYVJ0aBc1yzvPwtjZtojCpJgO/5co8FLLq2Hl7bpNVFHsTLT1gTUKQoCFmDs1hEX/3YqsuQAyoGITBGFu7/2qB+pEzK0EcoEaWJ7NVlKA9iN9VCNAUdKkaqNTkX5cbYFQVgc0mscuyToCFDYD3lpwrdK1zBxcYoh6B4uAGH5lcNuk3lQOvZPs5eQx2sE3OTdjFOY387k5J0E6tfIFKcHOGltiHWJst7NCS9F8tfkFyGKAEyU+XOuVzUHEp66uQL7nj4F2U+yxmpajxNy0ZGx75Kt/BEtS8n+cgSYU/8eySR/uWna0QwqjjuKTiL1ND6HuT7zzA9EcKocvcQm2AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
y = 2 when x = 1
![](data:image/png;base64,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)
- 1 = C
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The required solution of the differential equation.
Question 16.A homogeneous differential equation of the from
can be solved by making the substitution.
A. y = vx
B. v = yx
C. x = vy
D. x = v
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, we shall substitute,
x = vy
Question 17.A homogeneous differential equation of the from
can be solved by making the substitution.
A. (4x + 6y + 5)dy – (3y + 2x + 4)dx = 0
B. (xy)dx – (x3 + y3) dy = 0
C. (x3 + 2y2)dx + 2xy dy = 0
D. y2dx + (x2 – xy – y2)dy = 0
Answer:(A) (4x + 6y + 5)dy – (3y + 2x + 4)dx = 0
It cannot be homogeneous as we can see that (4x + 6y + 5) is not homogeneous.
(B) (xy)dx – (x3 + y3)dy = 0
It cannot be homogeneous as the xy which multiplies with dx and x3 + y3 which multiplies with dy are not of same degree.
(C) (x3 + 2y2)dx + 2xy dy = 0
Similarly, it cannot be homogeneous as the x3 + 2y2 which multiplies with dx and 2xy which multiplies with dy are not of same degree.
(D) y2dx + (x2 – xy – y2)dy = 0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Here, putting x = kx and y = ky
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= k0.f(x,y)
Therefore, the given differential equations is homogeneous.
In each of the question, show that the given differential equation is homogeneous and solve each of them.
(x2 + xy)dy = (x2 + y2)dx
Answer:
Here, putting x = kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
(x2 + xy)dy = (x2 + y2)dx
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating on both side,
- v - 2log|1 - v| = log|x| + logc
The required solution of the differential equation.
Question 2.
In each of the question, show that the given differential equation is homogeneous and solve each of them.
Answer:
Here, putting x = kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
v = logx + C
y = xlogx + Cx
The required solution of the differential equation.
Question 3.
In each of the question, show that the given differential equation is homogeneous and solve each of them.
(x – y)dy – (x + y)dx = 0
Answer:
(x - y)dy = (x + y)dx
Here, putting x = kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
(x - y)dy – (x + y)dx = 0
To make it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating both sides we get,
2vdv = dt
The required solution of the differential equation.
Question 4.
In each of the question, show that the given differential equation is homogeneous and solve each of them.
(x2 – y2)dx + 2xy dy = 0
Answer:
Here, putting x = kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating both sides, we get
Put 1 + v2 = t
2vdv = dt
log(t)
∴ log(1 + v2) = -logx + logC (∴ From (i) eq.)
The required solution of the differential equation.
Question 5.
In each of the question, show that the given differential equation is homogeneous and solve each of them.
Answer:
Here, putting x = kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating both sides, we get
The required solution of the differential equation.
Question 6.
In each of the question, show that the given differential equation is homogeneous and solve each of them.
Answer:
Here, putting x = kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating both sides, we get
The required solution of the differential equation.
Question 7.
In each of the question, show that the given differential equation is homogeneous and solve each of them.
Answer:
Here, putting x = kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating both sides, we get
log secv – logv = 2logkx
The required solution of the differential equation.
Question 8.
In each of the question, show that the given differential equation is homogeneous and solve each of them.
Answer:
Here, putting x= kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating both side, we get
log(cosecv – cotv) = -logx + logC
The required solution of the differential equation.
Question 9.
In each of the question, show that the given differential equation is homogeneous and solve each of them.
Answer:
Here, putting x = kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating both sides, we get
Put, logv – 1 = t
logt
log(logv - 1)
∴ log(logv - 1) – log(v) = log(x) + log(c) (From (i) eq.)
The required solution of the differential equation.
Question 10.
In each of the question, show that the given differential equation is homogeneous and solve each of them.
Answer:
Here, putting x = kx and y = ky
= k0f(x,y)
Therefore, the given differential equation is homogeneous.
To solve it we make the substitution.
x = vy
Differentiation eq. with respect to x, we get
Integrating both sides, we get
Put ev + v = t
(ev + 1)dv = dt
logt
log(ev + v)
∴ log(ev + v) = - logy + logC (∴ From (i) eq.)
Multiply by y on both side, we get
yex/y + x = C
x + yex/y = C
The required solution of the differential equation.
Question 11.
For each of the differential equations in question, find the particular solution satisfying the given condition:
(x + y)dy + (x – y) dx = 0; y = 1 when x = 1
Answer:
(x + y)dy + (x - y)dx = 0
Here, putting x= kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
(x + y)dy + (x - y)dx = 0
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating both sides, we get
y = 1 when x = 1
The required solution of the differential equation.
Question 12.
For each of the differential equations in question, find the particular solution satisfying the given condition:
x2dy + (xy + y2)dx = 0; y = 1 when x = 1
Answer:
x2dy + (xy + y2)dx = 0
Here, putting x = kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
x2dy + (xy + y2)dx = 0
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating both sides, we get
y = 1 when x = 1
3x2y = y + 2x
y + 2x = 3x2y
The required solution of the differential equation.
Question 13.
For each of the differential equations in question, find the particular solution satisfying the given condition:
Answer:
Here, putting x = kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating both sides, we get
∫cosec2vdv = - logx – logC
-cot v = - logx – logC
cot v = logx + logC
1 = C
e1 = C
The required solution of the differential equation.
Question 14.
For each of the differential equations in question, find the particular solution satisfying the given condition:y = 0 when x = 1
Answer:
Here, putting x= kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating both sides, we get
- cosv = - logx + C
y = 0 when x = 1
- 1 = C
The required solution of the differential equation.
Question 15.
For each of the differential equations in question, find the particular solution satisfying the given condition:
Answer:
Here, putting x = kx and y = ky
= k0.f(x,y)
Therefore, the given differential equation is homogeneous.
To solve it we make the substitution.
y = vx
Differentiating eq. with respect to x, we get
Integrating both sides, we get
y = 2 when x = 1
- 1 = C
The required solution of the differential equation.
Question 16.
A homogeneous differential equation of the from can be solved by making the substitution.
A. y = vx
B. v = yx
C. x = vy
D. x = v
Answer:
Therefore, we shall substitute,
x = vy
Question 17.
A homogeneous differential equation of the from can be solved by making the substitution.
A. (4x + 6y + 5)dy – (3y + 2x + 4)dx = 0
B. (xy)dx – (x3 + y3) dy = 0
C. (x3 + 2y2)dx + 2xy dy = 0
D. y2dx + (x2 – xy – y2)dy = 0
Answer:
(A) (4x + 6y + 5)dy – (3y + 2x + 4)dx = 0
It cannot be homogeneous as we can see that (4x + 6y + 5) is not homogeneous.
(B) (xy)dx – (x3 + y3)dy = 0
It cannot be homogeneous as the xy which multiplies with dx and x3 + y3 which multiplies with dy are not of same degree.
(C) (x3 + 2y2)dx + 2xy dy = 0
Similarly, it cannot be homogeneous as the x3 + 2y2 which multiplies with dx and 2xy which multiplies with dy are not of same degree.
(D) y2dx + (x2 – xy – y2)dy = 0
Here, putting x = kx and y = ky
= k0.f(x,y)
Therefore, the given differential equations is homogeneous.
Exercise 9.6
Question 1.For each of the differential equations given in question, find the general solution:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
This is equation in the form of
(where, p = 2 and Q = sinx)
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
--------(1)
Let I = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAF4AAAAYCAMAAABEDbotAAAAAXNSR0IArs4c6QAAAKJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmY6OmaQOma2OpC8OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkLbbkLb/kNv/tmYAtmY6ttvbttv/tv//25A625Bm27Zm27aQ29uQ29u22//b2////7Zm/7aQ/9uQ/9u2//+2///baT03hAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB0UlEQVRIS+2UW1PCMBCFs1UggoigeIMCtigBL62l+f9/zbNJSUvboQyOL4556Eyb3S9nz24qxB9Ya0nT3ysj7nz6J+E/xkSDJl36NLYQ6c1CbLyggZ9cNEUcAHDyRnpBItuRCYupZHRcFRAf3Qs1ZGQ7UgNLZ/6+0epsVZZ3NH5jrVfnTw6RyH18WMU3tWu3r7LGrruZNdg4gN/OiLxBZPzjByYWgJAI1Rc8TcbksdwYmzG0fj1rv/XKaIxSH+qzDFahfade++1IzyHEKEhkLxCKnQzRncJ8JXIi3hCRjghrKtayo0MoQMZE6HmeUcKnIzQquXR4cwY+8bGmh3aFIGm/U3WQN2xr84yCerGk3gtnZeoRZ07Eez+/PvaTQe2vXSy31mUU8WLTpRbmqIwXqjC81hOqDkQR7zL28AKmouoyPr1/yGmWUrOKeJfBNmWxbLupuoTXsyAdOa9rbbdd5BjrvcsoRHPnt2PEmBDz4NbqEHJjGqIR5oyYMCDr/CK5QpS3ENs7pGUZ9kxX63Ymia5YPObNPNjna588qCca7krAdfAeK50Faolb8y5pkmUwPh2d9DOut7/69Uc/zIZD0tuVqrkfx0prioPB8Pp/1TnwDSkxNPZH9vNCAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
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![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, putting the value of I in (1), we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAP0AAAAtCAMAAABS1Z3hAAAAAXNSR0IArs4c6QAAAKtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpC8OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkLbbkLb/kNv/tmYAtmY6tmZmtpA6tpBmttvbttv/tv//25A625Bm27Zm27aQ29u22////7Zm/7aQ/9uQ/9u2//+2///byGRl4AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADoklEQVRoQ+1afV/aMBBOOtHO6WTOzrlNikrda8DRleb7f7I9d2lLCg2ItP7AeH8UKL3LveeegBCv9OqBLjzw91LKfheC90Fm/ulWTILRPqjakY7Z25GYhMEoCw+nHS2xw2LVBZRLD6eq76HxE1P26uDbDkeoK9VU0fPGxx7lfXYpAwp2CuPTgRD/7nTc+92Vi3dMbhZei4dwIPJIggZiHB7pRPoS/QSG6vhox2LyTOrkEfV5coGPZBJevvnjo/Eod4q9r+RtyZuAp/Ja6PGLm28ei9nGoQy+vrSm1wpm0z9Pn6EdqrNOhqttMZuOz58lKbJ3mDFbpy0xm455P9APOPfo2cg/xUzYKlGYnETLB2cbh2FbzJaaKSBBX5xF9kTQuvVCWeOWjmu+VfJ8qn8sn7uMV0dgW8xWboYJjcGq7XDXQ20Hv259FtLyeriUHBnAiSEdSwCTDN17JFrDbHPxvDFirdkQOdifpoSG6ILtok+5gbUV3WsmKCQDfEmvp9TfCjE2X1FkhTG2JOU8bZt/o2NKnZSMbw2zkbSKEmQ+LaJvsBD7JQtPRiYlEjy4kK6WH0ihWTRgxWaxLcbm4wQrQ2lZr4nDUDGU82RuqFJQ0UMKIlZiNrh7kdHdbWzr2V6eirPTynp2AW6RV7jBNlKJoNg+klOJsfksnFVzZB65AdjcMySWimNTzGZ5svaWSn3e6IrEvJcnv8jCIvalQ/D5PSqAacm/JZAwr3wtxdT4jJEN7I+xnloUziPL9GgDs81jX4GBybHsIccWrRfu4qxglLHeSCrEwMFV7rpi/7jMp0glVTCcWfi0zE+Oqt32gVrwovX51RenuxtiD+1YjLD4XHXPTaWZbJfn0YcrhKVFzFb1fN6M0wsuee4rC9aj4vLIdTpUKmT6MnFWYiw+G2jXG2gqOWdW7XimZLjxtIfZSr15L9YJrKeufQkzefvjC3U9+gqfXG1PEYj8PuInZuSCUozN597vdRyAf+20U0xmOJZ8OmYb17KsmvW4XqD7MJQSIyfmCjngC21CH2PsPPTGZT4U6t1StYODfiYpxGBGmfOtmPX0PfgO1qHQdKuTSTLG2kC50uwRxNVJ2rm/cs5fuwTKoqEy1rJZD9QGu/L+nmA8Hfdvtgq9aWRLtB/4HjW0OQSs2dps/SbZs8/P+m79CVqzr39e0ENCYH7/d4MQm7fknldfuEtoWjVo3UciKMUnL17S7DMG3W5+VfDSn69Gt+mB/626d5zqy0n2AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
Therefore, the required general solution of the given differential equation is
![](data:image/png;base64,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)
Question 2.For each of the differential equations given in question, find the general solution:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
This is equation in the form of
(where, p = 3 and Q =
)
Now, I.F. = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJoAAAAZCAMAAAD+BcdTAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjqQZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkLb/kNv/tmYAtmY6tpA6ttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///bpU16wQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABtklEQVRYR+2V23KCMBCGN2hVaqu0tio9QdVgSTG8/9s1J1IkwGyGcdoL94qZsN/+e0oArja0ApRM8kGMwYCu6MWCPw2SNhjQWRe6bB5RMvOpogsAT0JXtHTrnKRe0ipANifB2rA8CHW3cyVl7EqjPtIqQPnyDscg0XQ84cztXBqPlDRGpjuV9GlFiGjoMQySIsTshwEoanErpHkTlJsJWBdXkelIHAdJGc/y8lVUjU1yusCsR03a6W0D4E9Qbi0BrTRRIfGt8lbtoONqcPp34ldaSsYihjdBu7UE7JSWzTHtBJmP1f4VbmvSsASQbgDV78XKrBN/0JNLR3ugQcKjRX6Kpzl8f5Txzb67YA5AggoRA0swgMrNBizCjdHKHw9ampj+QIhnIbnfkXUWzsq055lwAZCFOlscwQKMmwkIkIpmiXmVS3WnZ532tS8VurVVt4ML6J3IPkDzxpBPgMCzLTOPwSdusgyHRw6gf1mapxbgHqgSjA72ZWbyMsMbj5oAvK/60wLaNeNpTj900njDA/SYDbALAhjZQJnh7tTWDC4IkCv7jHmHukr794ABTb+6XivwryvwAziWNI6klP8TAAAAAElFTkSuQmCC)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPAAAAArCAMAAABxcvVMAAAAAXNSR0IArs4c6QAAAKtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmaQZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkLb/kNv/tmYAtmY6tpA6tpBmttv/tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/7aQ/9uQ/9u2//+2///bTS8W6wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADqElEQVRoQ+1YiVLbMBC1XJq4pYUUDKVHcIDG9JChxHWs//+y7kq2JfmQZCceaB1NZ+jA7tO+vWXPO5yDB4Z6gN0Tcj5U+V/Ui889+Dedk739NR2yyDSebSZFmEVTSmcIbRZMjHBKrieV0R6dHGF/Pa0Ix/8D4eSY+F/d4saiV9Yx/HRJyMINzusj6whpF2OrO+/BMXAOhPMLd7g+snYifSSyN2vvIfDXWWDeK/JQiTD78b4j3K5waKIqS09/SqtTMl6D3N4u4aJ0tqEL8x6Vh9IhLDrrEqY4rB3gkJ0mm71Tpp7LCHyE+vFP+y5/MTlCwh49spWyQri2c+Uh4QcNfhAlbIdrymK8y5MFzZnPIu13lJxt2PdmPSa2beGRYyfHtkVZSfm0s33RomVJuPxK5P62+CnztiFLpQl2wlkwByi2aoxKqcoiQmabLCDgleySd+f8Av6LhP98Y9FrpYhamoAkzCK8C2tQoCi1B3xTHQ5wkXEeFq6olNKGbBFiQCUnYFSM5lJZzXqEaWevlX9hEbowRb7B0uORTYA+mJwEcxYDfAvP6lcpKWgKFyHfEqWUEbl9rcPxW/NQKourW2RFpSAqu8H6wMGvsNQIKzOjrChRV3gqX1BMRQpX40OvilOdJXi2rogikjC6DI8RRaIC46fLeelMk1KMXuGPUN600FW8r4mjEVZbaJ2AdAaGBrM+DxGm5/O2QdgZBWqpSh6jElokBEQSZcFJUQiNILgRxqDCxCjSidg3J9V3kjBPFCxLngh2FJ2wQalO2NMKdUBKe2BqDK4TXuw6HSlNq+93RUqbUWopXYbYqFQnnF99UpcdbSx1b/aql/LwA06HzvI1uAFeh6WXCsKOKGXTEkVsVMIaFgKihldr2exqNQwSaI5xLPE+I6xOCXTCxLZq6PQl4bJLO6GI0SDbtEFJRJ/6d972IxBm+IVU8Go2LXCMDxysi0e5MuA4+tJvK5Pv/ypKLijbz+KarPjJJ2H71WIOw8dvf/E7IMsIxgv2iZJxbdNi9wEhRzYSaTkOTcnb/rdY7vPdm1Z/WEVD2bR2wqlSYrVuSXlnaKVLjPP9Ut2lna0yCbJocTM8wHzvkeXU+VoabKr2WhqMoirCOOz9mJL6+vu/+z081FTtPTwUZJ96RbfdJ+TLxlIn4su2dE/Wue5Ve7ru+WHanuTPb9WIFow0e0e0eEfofW8FO5ozunptrxv9vue9AD594TN6OgdW+EnxnU5kD0xfogf+AlcTcXQrGeA9AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
⇒ ye3x = ex + C
⇒ y = e-2x + Ce-3x
Therefore, the required general solution of the given differential equation is y = e-2x + Ce-3x
Question 3.For each of the differential equations given in question, find the general solution:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
This is equation in the form of
(where, p =
and Q =
)
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the required general solution of the given differential equation is
.
Question 4.For each of the differential equations given in question, find the general solution:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
This is equation in the form of
(where, p = secx and Q = tanx)
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAW4AAAArCAMAAABvnpLWAAAAAXNSR0IArs4c6QAAALFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpC8OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkLyQkLbbkNv/tmYAtmY6tmZmtpA6tpBmttvbttv/tv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/9uQ/9u2//+2///bkfuZegAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAD70lEQVRoQ+2aC3ObMAzHbbq27NFH1pW13aOhXZN0j9BlCwR//w82SRAQgdhh1yjbxb5eyhFbln/If8smSvniCXgCOydgJlpf7tyLvXFgfKngzxcZAtmrqUxHvhckMD6aexBiBEzshUQMtlJZ6HEL4k71ULC3ve8q8bglYyAJRpLd7XtfY49bMARMfODTbjnee4bbfDvdWnQl5z+czy2P+ka3mMepfvZV3MQXW9zTZW+cSV4e9dxUCnq8SY46u9I6ON+QYd8tnYkZv+ylM6dwV2ngdldXkh5nYUe0NBCoRF/Mzdf2av/UGWdpz5nc6CvdIKdIXLGbhazGBhYlPXbjzsJjkEtz14q7uqmJtT6aZ6EORibG2kot7mBGDGBGZFc6+Ix34EIHwzHWTJiEcdxwSq31wdT8fK31QClUuqcQr3grZ7xy3BtYFPMYAZxBdFsRqPWbhvobE2NAYSSVzwBvmHu4mYW3aoYzCC8WEXQGlTjidnTD+OeGcucsPBnB5ILWrJVz7qe6eOJUKLqtFqU8RgDmfmUw5GMTR6UOeYTRx0s19xOslMAwy8mbR3BKlJ3O6SyU4md5KIoPImFHSGvEhCCUH1CbtxovaUKYlKWhQW3cNKh1FqU8JgC0VNoQ2Nb5OsPFsZDeLLVyok++wxgJO6IuLophn4E8UGnxqpW2hlO05K0c4u3E3bQo5PGyU8BtQ7AZbgzgFDFQmGMB/T2cAm6Kv4NpjXtFnTqi28yuQ1J3Yl62rJXL9fKgA7fNopDHHLcFAdujWcQEMY/xwbFMYAZrbEW5xp3ffODJS7d2s6lftGStKtxrxCThb4Ur7V5rUchjjtuGYP2BD19E8+jtDcZ16TzKNmrIctkv9BsLSE4esbWsjZtMrIgJb1VpdylJq//auK0WhTwuABTabUVAwWJNBAsdJnEu13nKQ65w6dSwHj9BJpjgxePI4BvylMVfc5sD72Eev8DH4iPqCDmH72Z4KyZL3bwbuLG13aKUx0nwoBbXMCI7AhMHAMq5zSk3C2UUL+5AfWkjColz8Km8OHyADD2A6Ga/AmluqSaQmitIlge/Iz2ERF4PsfI73sqZdzffLjgtSnkMv30JBr9CfetAYCYw7BeEzFLSUiD67tHsVju+de4qxz2PgXbvcU8GoDSV2Dh3IT1tr1Z3BjdtifqU3Xvcx1sU9nhwX61+gudr3W72xQ3u7/gMszfuQqeLInZ63O3lX7xd2LHHPWn/W9WL8xtfhAg0snqhPve4G2devsdstjD0zhP8LfTjTRKBrafRnjMn4NwGeVzPSKB5KPCMhr2pFoH8/YhO3X0RIQDnWZ62CGnfiSfgCfw3BP4A3CjOr+zJTsgAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ y(secx + tanx) = secx + tanx – x+ C
Therefore, the required general solution of the given differential equation is
y(secx + tanx) = secx + tanx – x+ C.
Question 5.For each of the differential equations given in question, find the general solution:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p =
and Q =
)
Now, I.F. = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMMAAAAaCAMAAAA9kBh/AAAAAXNSR0IArs4c6QAAAKtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjqQOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjpmZjqQZmaQZma2ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkLb/kNv/tmYAtmY6tmaQtpA6tpBmttv/tv//25A625Bm27Zm27aQ29u229vb2////7Zm/7aQ/9uQ/9u2//+2///bJDQfugAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACZ0lEQVRYR+2WfZOaMBDGN1g5ub4JPfqmcO0dtL5E28LF5Pt/sm7ARGoAl6Od6R/ujKMzJr/ss/ssAeAa/18FOPOLUVl1A9Q9Y9PtKDhps5jLD6M09ADk3UbGISmNUYu4cwZnsyFEF9DcLd+f92EgnpJKnjir8kEaWgAnovjo9ngAnpMyUamrgbbzmGgbwGooPxVicV6jAXiaXBlVGkp2s2IeHneIGUMv7QMvEwFl2I8AgF3A2AIkjvEcQH1lzN/KCGFhEzYMv8LtfnFAVogZ+uvASwzsD7/WGoBPMth7mUpnhfqCHSz9gs8ps240iNsEeKjSEA5RojHipZ0ECxuK132Q0RK+YTW59wA5fruZmRQ4/ou/xW2GerQL+QvHBK3jZQAq9T4XILAZWLUK0wgDG4o3XsK8QWeoP25mnRp2ryhO0mUyA/Uz9uoaAJxrMLCTBhq+0rB/rR110mBhceV+TOFdXTE+2WC3MhnNi0N6U8DTo0qnm+4HmzgHYKnQSDJCL64Rg1/frRUtjIo3dPTOr9LLgONNafpgYCJYwo9AF1De1a7laAGcGCgD9nbFFrtgpvKeC9wFPKGL8CR9+hIbEbOptVMDRsNbugi8hUzZdM1m6FL9CY8wqGYD50t3/U1drcppXZFrk1eBNanCBXTvvvCPC2/Qu/fKSF/OmEeZlMdrWk8+PVoA9M0XV1p638rquc3YZGtf10p9KdDDBdD3Xl5p6f0ahr2IOe2uK/V3wvUSiV6PwogYDeg9m0Yv2RLUjnaJtR43GtArgkbHlxt9rT4/RgN6j/639Oervu68VuBagWsFuivwG3OTUjMC8AfqAAAAAElFTkSuQmCC)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
-----------(1)
Now, Let t = tanx
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ sec2xdx = dt
Thus, the equation (1) becomes,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOYAAAArCAMAAABbwgTjAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkLb/kNu2kNv/tmYAtmY6tmZmtpA6tpBmttv/tv//25A625Bm27Zm27aQ29uQ29vb2////7Zm/7aQ/9uQ/9u2//+2///bKf8IpAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADA0lEQVRoQ+1Y/XuTMBDO4drGz2px6la2qaVuLmxaZOT//8/MhQJJFkhA1jraPO0PPO29d+/d5T4g5HiOHtiVB/gaYLErZfvTEy+I+Iz9ZK9ux04R+cXTzQHQ5NH4E1aEMaMHQTOF5QHkLGEHQjNYHUI0473SZLOBfOwA4tELS9scTLuLRDwUTQeQneZg2h00rwFgunm4wmkzhekNDZbkjgarjHZs5gVQs7Y8tERTEUrhSUsU+jMPz8la2MiCr3JWSacbNu86sjgCk4c2JyhC7oZzfwoQvOtqV+H5UpGgRtj2S9jJWUsWZC9tNfNfaWZU9FUdmkdqq2XwYcOvHxeyxKcfS+vu3siMq2gmr9vyL7XWTAdN+y1QhCRNHVqjmVG0lF88crEUlIdHSCOjUBuYnUKAIRNJ+hvB2eS2ovnnG48mP5viKbZGgPqiqUBtKaBddkNIPMJbYa0BrdFkjQ2p/oVHGB7FVxk9J/cyTWhwlkcwuYGZcAN+Fwmd8bilnmgu14BaaKaglHRDCB/5JZag5mgqlToP0dHqqTzA0P1Kl8JKwyOPZhJXcBWWZosvkEbTEJIrmixBzTTtJaxM1jK5MIGVvM5DXBd6boCqLRYgi2OELpWmIVQ8GnfTRPGjiZHDYro927jbBhPXPKG73BtIpylTpNRupVmUlLqI+iUtEVkb10IFss9xJK03EFPedhlCXjRJ80ysFqc8fP+5Hiq9rmWTF9Sk9QZSaRpCxaPjborfMTKtDUXOAfJveViUnRREcUvahoDmUON7gB/fOwKpNE3tOH09fMISJKGrzqg1FB4FwmTneJDKq1DSJIlool/6DU6ivQXLrkD6Vq1rFy9wg/kvijwRunKwPgXxNQU4cdmcenQPn8va8z/xk07m28ZysbIkdU+D+4ntYqvm0fxyv8FsKZT93GaTEjNtzwVmKBvsW/VQ6P8NTjFUj/5UhXncTL2npefthnr3fd48HNYXw8noD75/Gf/Rx7Zx8s0/rpRVd5wc5brQ+qJ6tLyPxI4eKD3wFxviV3zxo1f5AAAAAElFTkSuQmCC)
⇒ tetanx = (t – 1)et + C
⇒ tetanx = (tanx – 1)etanx + C
⇒ y = (tanx -1) + C e-tanx
Therefore, the required general solution of the given differential equation is
y = (tanx -1) + C e-tanx.
Question 6.For each of the differential equations given in question, find the general solution:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p =
and Q =xlogx)
Now, I.F. = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASMAAAAfCAMAAACbMVt9AAAAAXNSR0IArs4c6QAAALdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjqQOmY6OmaQOma2OpDbZgAAZgA6ZgBmZjoAZjpmZjqQZmYAZmaQZma2ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAnXs6kGY6kLZmnbZmkLbbkLb/kNv/tmYAtmY6tpA6ttv/tv+2tv//25A625Bm27Zm27aQ29u22/+22////7Zm/7aQ/9uQ/9u2//+2///bsV+btQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADgklEQVRoQ+1Ya1fbMAy1mwHNxjZKeGwwaNi7KQy3bPGC/f9/12Q7cR07cR08OGMn+sDhkEi6upauFRAabWRgZOCZMsCvMN65fabgnwg2O75hJ4dPlOz5pmGnYx/1nh7BuyVC1Rn8+F9Nlfhwq2bsXYnoeVldPDzI43rGqqUqMcKIkCGWYYyVHhE8jYj2KK6xailLbNnAIou55V88jKO7s0+97K7OA47RiztKLZ0SERpUJM9tjsgQjn6eYDyT7Qdn1Z+4ehNAkgd3qFpqOOa5uyUC3CFFsszk6B5Khm5Yp5NFlQboHDv6htaTBUi+IMGT2G13dyr73UPVUsNpBa9LpHjvGk9AdIcWCUpkcMTzack/AsV0tySzgKMXWKqXwJHkQBQpAByA5zoFtpMvDdnVq+2bhev+Vbn/2qjldr2TcLo4QiRZiAMdXqTiCOoBq8sVbUheBN9ygh7VzlCkAHCfTYH6C0ShwRqy+ZWNvLOP+ty3U6PfcDu2aQMCgwG/qzMdUmSrjzbuq/2ASZO41kKONEcqQnIrOZK/K7K7NMG5bKb1Ibnu4RxJOD19ZHMUWiQ7Mg6YZbPyPt8r0e/vPN+5CUFGJCaLIwBDX8vZRzWOQRy57iFI5DsKjrTqRCFATYkkuUFksgguchPg2BQKmuKDa3yxSqe88Oym2htRwETnNUfLvZJlMGs5zFodtiHbN2tNNI+7l6MCA1SCMYx7DUdSlF6iu1RIQIMFXsET8VJgkTpA0K1sQTTSCzUV4IQKULGGQsEY1J7nAtCGbI9mN9F87v4+KkD4ZKPKVVjAASugG0HeBFv14iH0qNeA6NrENW0GoHMqd1CCkx9Z87Adx3U20m8a27q3JK7VBpRW0n4sZlrb3XjWgYfn0EfWLs0y8QcgqikRoWWoxMpsOkDzsceLKV9e+k9LP9Xe5vuGEEjSQc34Ui9qnm7tjtZ23wasSt/aUq1aCie3+nsW+nTY4lgH0Ml53t1FXeh0+tbDuzNT4/kSMog9SZrvW6QzWtt9G0VwJA56RX24Oe3pBuCf93tiur09NH0LaAAWf2Eds4bY6fvE2lGVFEWYEwCuHZaFzlp0+hbyvxBNorcpofgS8VXwJuyyaQco8CHFwdMWnb4FKDoaL+Slas/BKsWTD4HfU539FhcgztsGFBkN1gyxGDb//4oYr9F1ZGBkYGRgZGBk4B9l4A+jNoJB7k3Z2gAAAABJRU5ErkJggg==)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
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Therefore, the required general solution of the given differential equation
![](data:image/png;base64,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)
Question 7.For each of the differential equations given in question, find the general solution:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p =
and Q =
)
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
------------------(1)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKwAAAAtCAMAAAAEA3CXAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjqQOmY6OmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6nGYAkGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29uQ29u22//b2////7Zm/7aQ/9uQ/9u2//+2///bVffNAAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADL0lEQVRYR+2YC3PbIAzHId1a79Gmbrot9R6103nuHp5jvv+HmwQ2BtuAeiFOd1ddL+cmgH/8EUKIsRd7XgqUfJU/LyIfTR0T9vH2uDM/BFbsOL8elNhv+Nn3KMvUptwauR/0ENjimsFfb83VQxmALYDh/E94Pm0630rBNhvO3z/Ag9glMCA//4KjlpzfeUZu3o51DMEy5sKwX+OFbZIt22coS3GWsx+IWMAkROZjZcVk+ovAFheobnIHePDUpvAhMlDWcMipwiKb/LwEbJvia/Gzg8V/m+Ry7fWuJjkhrATF1a+Up5aBoFZPHXpZZUFazldbZG0/fPTv7fI0sOifymeZdlPxOZe+67YZ5Y+uLOpTQ2zfI7HeNAIDKH7tNvSYkYU858DQVcOqwzsriK5rjNYQWsHWvzP4Gs8RN62Qoc4w8Q1D9KtP3l15WJwdReMN7q2/6BAhm8CGOsjfY8Iqp1OhLGBtSkkEYOWsiTthxSMcoq87x6JNSXqp2FEwaAOC39NgCw6HaD9/4tjVO/C7K8wSRjY5eokDyhgzmLOXPET7aEgc27n2E9gmoeRPKiASYGWTfhlOAYu53CXAQmbYp3B+jKLzv+iwNfceGaAT5nLiq53CdRiY2Cozg7VehRPAyhRSruyQwvkwhjTOMSU9Q8eDdKVZHSxlJ711AFRq6RTOAysTKWXLK6uidbe0/UHscYPiQl94osOW3sRhpKxO4dwYpXQanN8RlA3CqmVVPqtTOCds8yaHdt39c3llIXu/Z/tbgDVSOPehoNz+aMrOZzu/ILiq+xCUFVbrnwnfGikcTTNaK+cBNv2hmL+mtzf3rHJfiGgYtFZPgXUTSf+bNxoGrVUcWM8VnoYRbmWc4ARmO/e2+laeK3wYgxi6CEWYYRbqkqnN6Fv6yg3RYMNFGINudPUd+tbAWjuvRdFgw0UYC1aFxN76019WK931vHiwwSKMBTdSL3wNJ3oj8bgNFmEM2Hp0TyP2jaZsuAhjwMq8w9hvwQKOahsLllCEMYOB5QXkvlRYnSbMR1EoyYWKMH3H9iavLWGpfTGLJ10zCYGe2gR2/NKvpKK9tPt/FPgHCLRpUutj+T4AAAAASUVORK5CYII=)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAH0AAAArCAMAAABiiXf9AAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjqQOmY6OmaQOma2OpDbZgAAZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6nGYAkGY6kNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ2//b2////7Zm/7aQ/9uQ/9u2//+2///b9YrvdgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACHUlEQVRYR+2Y21rCMAyAW1CYB3B4gKlUnKBzrO//fCbt6MpcT1CHF/RiX+Fr8zdJkywj5DxCLZDTwTJ0T8T1xYH0zX2MUx9G387o8D3ABlVKKb37veEgejlZ5YLOQOjoy32MKu1eJenljNKbFUz4WwIC6egFpeaULkySJZ0Qk9z9fVZ6mczJNkN5bLgkH8hkcCqeGeEkIp2NUf9kATyYVSk8eAa6d3hqp1M8epUiBp81HX+Wye3U4tDYdEFGg6+lt3NrLMamCwvwjNLBXFynh0dbUMWjo4+l34lyNX9eCv//8Z1HMxeQCbZ4BJ7VF40zmODfRnrtl6MirgBDQ8CvIcqnmDUgxGFMPzP4GxNUN56/Yla4eDo63ltZYYYX7ht94DmO0r3FkHdJRqDfMNL5BvLnZV2K/M4oPM3fAiqIUS6jkD/ToHxM1tfg0AlmfM9hpmPA5HWh8NPdE6kts8stTkpnYZYPVr7WHQu9HHp+FglMJE5DfQ/GtTbY5KrctaO3z6iObJgIlnmTVStRtnrRvcvybKzeufq3fI6lq5B5q3d6eQV5ThSrU9BrX5yIvhcbvVv+TPeOd7gm9vZF2VJf6edRj1Wu9qXxpLbSQ65fxLnal4aurYxGd7Uv2i1uGp14dEf7oseQanTi0R3ti/5GoxqdaHRX+6L5vWl0YtGd7Yui6yt96d1fThqR9vZFMzt0mrtGx/fLybFvUP94/w/TOUH0fpL8AwAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Now, substituting the value in (1), we get,
![](data:image/png;base64,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)
Therefore, the required general solution of the given differential equation is
![](data:image/png;base64,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)
Question 8.For each of the differential equations given in question, find the general solution:
(1 + x2)dy + 2xy dx = cot x dx (x ≠ 0)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p =
and Q =
)
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the required general solution of the given differential equation is
![](data:image/png;base64,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)
Question 9.For each of the differential equations given in question, find the general solution:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p =
and Q = 1 )
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
x(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ y(xsinx) = -xcosx + sinx + C
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Therefore, the required general solution of the given differential equation is
.
Question 10.For each of the differential equations given in question, find the general solution:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGwAAAAuCAMAAAD6KAyYAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOmY6OmaQOma2OpCQOpDbZgAAZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm27aQ29uQ2////7Zm/9uQ/9u2//+2///b5V/NIQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB1ElEQVRYR+2X8VeCMBDHd5aiFGrptKSicsj+/7+wGxGMgdtuPPH1ih98T/2O793tbnxg7K9eAmA7Xu4izOxjdQiIMY8CMjst4eZtJLN88ZqRzfIlwDzaSg4wPeYRTPxLQzbLow2Te9wzyadHxgTBi5HN0tJCNUi5NJsRdoFqViT3ePeyQdSH3PkXsQqPEJxmJvmMCZWn9zUgM1XHlDQCVDOVTrVnrEju1qTBoZqxbPLMTqvvEyQFtYH+V0ZpXXVb+QKT+DMq3QRlSOVTBAC3j/6xtZWC0vehJmWGuwOt7weZ8Xg/VmJ4WsGCMmPeib2Tpsn7tv3CoMdXsCd5QChORYIj0bqc45jWcqfUFYnk1fyaQeD3c2t7pH0/dZfXZq6oAv6/ahkv2iBGMUZtfXOofwURB3QPLgmrq+RBx95oZhURIxMgEWeW95mOQM+sAWpbbWsiZimeS9bKmIKW2AuoGyJW8sxGPKagHZkHUGuQim0yj60Npgm657QHUOtmiHUO+G4L2pl5ALVuVqwf7CxnCIwNdgO1RsQIV0ViYx5TYJh5AHVNxDLF5hAWJu4IzNZ1A/UPEW84vnOqR9G5fsRJMgSmGQmow466ZtU/UA+t4OWA2h7ZF7lCKPyfLQiBAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p = -1 and Q = y )
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
x(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
=> x = - y – 1 + Cey
=> x + y + 1 = Cey
Therefore, the required general solution of the given differential equation is
x + y + 1 = Cey.
Question 11.For each of the differential equations given in question, find the general solution:
y dx + (x – y2)dy = 0
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAXCAMAAAAiJm1nAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6Ojo6OjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZrbbZrb/kDoAkDo6kGYAkLbbkNv/tmYAtmY6tpBmttv/tv/btv//25A625Bm27Zm29uQ29u22////7Zm/9uQ/9u2//+2///b0D7TRgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACBklEQVRIS+1WaVPCMBBNEKscKipiOCpqRElt/v/fc7dprrZJU2acYRz3W4F97+3bgxLyH2ftwNcDpfPzVVjeb8lhtPt9gYLS5yQWuZ+8+z8srvr08dmrTUlm8llEmj7Jbo+NOvhdb2HFtVN8IlMDtMhS/JOsJeaQMn6uxzGmj6CINH3iotFcwlPkEcLHxvYYk/1OMkrHxyKjMN0FLOEU/MvxIx6ZRMkuwXY3VYA8ESjahasNrJkMQmsyuFk2ybAkgfKyFZEbnL8cniQLN7ouz6aWCwoRSnDg1GAYJo1AFIAbRiDHXnEwJK+EIgum2XGH+uvQOVgNhkmNb4YLl6PzlimIIJkeITRDrndQgioNXSiyaWyetD6d6str1+PAoTSHqRuhGh6tD4dJ+FngTPTYVlVXICq1NyxcQ58lD/YXu5SDaa5/5fLJbmi4vzq1T58D19AXQnD9KRc3S/BDLaWav/WuXOBTIHR/oaYqtSdcOJw/hymA4N+enFbXlo+25PsR9MkcnoX6sDNsep0aFejCqSYZJtyVLh7/PtfXVr7Q0fwzoysGxxAXPihQGYDRPtQtqXDkLJy6f5oJVzEBQURa2e2MAR2a6vx/6Arj5DAbeFwGRnVmT0htvOMkIEg23wy2D0f89jg81Xt/wT73k8N0zBIOWHuq9pO3oane+x8inko+sNt//+c/oCk5a2O+RFUAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p =
and Q = y )
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
x(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the required general solution of the given differential equation is
.
Question 12.For each of the differential equations given in question, find the general solution:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p =
and Q = 3y )
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
x(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ x = 3y2 + Cy
Therefore, the required general solution of the given differential equation is x = 3y2 + Cy.
Question 13.For each of the differential equations given in question, find a particular solution satisfying the given condition:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUkAAAA1CAYAAADMHRmmAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAfqSURBVHja7Z09cts4FMd5gBxgM2O4ywFkyGW6HZnJ5ACxOarWkyrDGU/qjOzOM7u+gLt0SZHaxabdxjdw4Rv4Dlk9UiAfQYAEIfBD0r/4NbJFEh/8472Hh6fo5uYmAgAAYAadAAAAEEkAAIBIAgAARBIAACYpkmkSvz8R0aNYfP6CjgMAQCQZnxfiSxRFvwmIJAAAImngOpEfIJIAAIgkRBIAACCSAACwlUiu0vM3tFGTxSLFyeNyKS+5SK5Wy9cyip5VrDI6in99ur19dXv76VV8FP3in6GjAQBjUtEzK/J5uVq9dhJJJYAiTlYkcvwG3JKkz3OhrF6cvn8Snfx7nq7eYIAAAGNSMdxCieSFjL7pVqDa4dbdbfW5TK4/8M/glgMApgCFCkuDL54JeXHHxbNNqwxm6foiUfSif9EWkyzcbnnxrbixED/idDXDAAEApiaYFa1aiyQ38JxE0iaGTRs3meUZiRcSxkxkRfwDsUgAwCStSm1fZRCR5NYnXG0AwFTh+pTvnYjHNq/XKpLKJHURySIwKk4e53PxHa42AGBq6O61t0iWqT3VnR6beNb+jrQfAMBEXW0VFuRaR4YfCWgi5cp5d7s4q70RPEr1OZPH93pOpEml4WofJhdS3PEJCMazlpL49IrCX22xtkNChQS58VdLDbIYgNaLpgt5mV80+n0sz+7T5PQjvQSncXJlsxTpRRnjJakliq7d/jhJ32NyQCSnhKqkVczRHvoq30TN3wOIZN1DNu21ZDq3Hg9bXnfYhzg+ux/a1S5XiHpyaB+ThO4nF+klJh7wcvU2izcJZl/Wnil3GfgTdGCGdrULF3+TKKqsytKENmfQb7tSI6QAQlgxuZiFn6MQyQmJJK8zGWqwu4itzaorYw1h3T/bqSMA2udNfS7aDm5AJPdIJF38+bEs0jw2Uxdun8Ic6hRRzaVnmfuVja31C0HWrS7cWUBdRE80eXl8io5JmcIUtWfdCHT18/7jgJX2UbGTdPlWyrJ9amNPROJJPYtPe33bbFu8plCEpbiX4T76abXqAl81PorsEcPc0z0nLpK0l6DCUbZ+r81f9j4POY57725PyW0vJo5lt8q3MIcpV7ScpPJZfS+JxUpfyXlAnTbCVGyqbdWn6xffZS9a/izlPfuEhxiK59n0bWbNGya9b3t92txk4Y9dhKUQhYaMENOizE+x8WuZPCT6/4VcfFVtUf3h0u90zbkQD+r/9P4achwhkgOKZO7GrC2eBvffpzBHo0jWJoFbMRAXl4tbC/TdLufjqyGRZowiYzi6lX3GLElbeMO3vdu2OcRYB+s/tbFoWLD5AlvzeCz9ZFoQ+FG7rv1uirHr9xhrHLfte4hko+Uj7triMT6FOdoKD9M1Ku5NB5G0Wb1190U+06mmoeJNFQFsSKsyWT7btDdkm8cswuIrkjYrs7DM2PX01DvXfm8tIbb536mM40Fakl1WCteO7SKyXQtzNImkEkeK01COaWiR5Pcf+lRTniOr4lXi6TRJPw4hkiHbPFYRFlPc0cUVL/uPnRDJnnv+kMcEc2EloXsn3v1jKmvY1u+2XXcXS3yscRxdJF0Fy5e+LUkaiC45jF0Lc9hEUk+k9nG3XUXy6OjovzF22GunrrTn7VMkQ7R5rCIsXTdumqxQem7qS96v2f94zrO2+09xHDsLXACt2ht3O9tAkNUdZecJ7FiYwyR+psnSh0gq6ye9Tv/okt4UOq7DTzfVN6YCu9uebQ4x1iH7z5YPqfqiadOu2BRcnc/UYQ0ubuS16G3pLJK2LJDN+zTWOCImGVAks8D1ptqw/kJLGf/t4kK7uAEm8XP9bJvJlu9Azr+riWw6h9qnJUQ7p7xvTC93aJHso81juXzNmzDN7VF9SMLOv6/6W8zLPvLp9yLGqaXxKat1iuN4UDHJECLJN0p8VpguhTn4xMjzx+K/0vTPt3zQSZgppUK5pGl6PlOTT01Ik6DaXlzb8/EiJH2mUhT3Zzlx+svNc99MaU9d2tu1zdn/zqOfoce6n/h7uYioedu6wWgRFVtVLp5249LvpnxG/XDIEOMIkexJJCsbCkbczPouhTny+GN15S2LgYgXiomWE0+80CaHaRdRxcYqz2uYcDxPTU/J6NpOb5GMk2wx4MUZVNtNZ+cpkd63vV3bXO3b9n4YqwhLbUF3PIBhy/k1pWb5zrPsKC87LEH5kLWk+57HESI5YcYqzAHCY6sHiLEGEMmBY6BggoudQ3UmjDXoVSTVmc19mGR9FOYA46Biw7ZEd4w1GEQk+UTbB5HsozAHwFiDXfA46gW62+ZFp8mGMmEAgN0VSMqlLlMCS8Fs9jAgkgCAvceU76uEsy0VCyIJADho69JbJCu+OxVaXZZFG/apoCYA4HDhtVE7iWRRHWTz2zFcMGvFV3e8oCYA4PCgk3CZps3jn215s3Z11bLoTcU+QxZGBQCAvjGeImr4GQqjSNoO49tikrteUBMAcJhi2VQgu1EkbWLYtHGzywU1AQCHi0tVomAiOVYxWAAA2AbbL6u2iqS+42N1twMWRgUAgKHJ9lsavOC6+WmpVmwSz30oqAkAOFxcaozalZXFGGu/b7L+XFmOu15QEwCw/xS/J56d1U5nSiCpvqjpVw1aRZIoC8nmRTjzArfi5TROrkg496WgJgDgMCzGiqFHuiXmD00/k9wqkgAAACCSAAAAkQQAAF/+B7j64IThiXILAAAAAElFTkSuQmCC)
Answer:It is given that ![](data:image/png;base64,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)
This is equation in the form of
(where, p = 2tanx and Q =sinx )
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
----------------(1)
Now, it is given that y = 0 at x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAeCAMAAADJnMQBAAAAAXNSR0IArs4c6QAAAGZQTFRFAAAAAAAAAAA6ADo6AGa2OgAAOgA6OjoAOmaQZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmZmttv/25A625Bm27Zm27aQ2////7Zm/9uQ/9u2/9vb//+2///bCtCGjgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAZklEQVQoU5WOSRaAIAxDA4qzOIDzBPe/pAhsfLIxq/+Spi0QkCqJFccwSo4ptjMB0s2TXhugco6dZYZ+y90y+t18F+aU0OqxdNtjocKnR+Lp7GpnSRJ5AlbGjaMKgcMSZua3fL+5AZnXBDVGR3G5AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
⇒ 0 = 2 + C
⇒ C = -2
Now, Substituting the value of C = -2 in (1), we get,
![](data:image/png;base64,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)
⇒ y = cosx – 2cos2x
Therefore, the required general solution of the given differential equation is
y = cosx – 2cos2x.
Question 14.For each of the differential equations given in question, find a particular solution satisfying the given condition:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWwAAAA3CAMAAAAff4ALAAAAAXNSR0ICQMB9xQAAAMBQTFRFgYGBiIhmZoiIboBuf39dXX9/f25ZWW5/WX9Zf25Ibm5ZSG5/bl1dSG5bbltISFtugFszM1uAbls1NVtuW0hIXV0zbkYzM0ZuW0g1NUhbbEYdHUZsRkYzRjNGWzMzHUhbMzNbMzNISR1JWjMdHTNaRzMeHjNHMx5HWjMAADNaNDQdHTQ0NB00SB0dHR1ISDQANB0dHR00MgAySBwAABxIHBwcMh0AAB0yHQAyHR0AAB0dMwAAAAAzHQAAAAAdAAAA5i7CSQAAAD90Uk5TAAwMGBkZJSUxMTExPUpKSkxMWFhjZGRkcXFzc319f39/i4yNjZqamqGhqKioqamtt7e7vLzFysrK2dnd3ezskMdkRAAABM9JREFUeNrtmmt/2zQUxo88OcEzNHZgzZCgG7Tr7LJugNOE4sv5/t+KF1Z8la+R8guJnpdueyr/dXx09EgARkZGRkZGRlp1s+MGwonEEE8O++0vr1c6wd7JYXv/vqKBfdH/08C+Ji12uH+HHGiMuLVIhFvLwNYkGn+wFjvkAIs4sQHo30uT2boUbC0AhhwAGPoAjJsyoktOyosXp3EI5Nk1sNW8HobyVxYvHmSu89UysFUk8YPI407YTspPVkUuHPb3FgBrthoehuWLk2j/5BrYyt6wCZvGiX1ADuCdqu+7CtjtKsFway0+YU6ZRKcD4LUXkPPX7WZCU/3ZAgC4/bH69yk+rrOf84y+P1UVcVJEzfYXeZ/6iiM+P1jjHbQ7O/+jqKNaiMno2WzifnVmyXazw700RQLEo2HXyd7zCQ4aK/55EI6sMu1sRP+sWHvZCm7kKcyOHmqdrOguxi02HgfwfJHbXDI2TOy+jyj6YMEi6v+l07sNHACYdExMQV5UyNLYl6zsHfnJEEucXnt4Xrpf9uW6s8mRZ+4ZwWaZW+yDNcM+zGgXbBojIqcxtgDJhzemsASJXYn7gx6vMDdrhp1IsfrQOAQgEWJie4gYAokwsRn66xQfLAAA8glxvwTy/tW/2Yln3XRksIuFrjOzSYBbC7xk2THK6bBJFNbi6vEKBezB6DQWLevWAiBB5gLQKHMByK9LYPi4EulN/1zBIk7sAMtnPXRksGkcwkAZIRH6UnMpGFd6W7Cdf+x63IZXyLAQV0K9Gr0V3EnDfDSJXXyuDHm+PjH0D8+Cww8qzyR02oMvyYp/1VuzaZw8yUpXMK70tmDf+424er3Cgeh12HmCBxjmGwYBVlQYRMSw8qyXjiSz27ArU+MXvy4tGDls7FBHrCr8Iu6AV4iT1RxnT3TxbYtqAixzwflrl9jkd6sKO+9Zii+lhD1oUEzLbPBepN/zvMz2Nu24er3C/ujVBVIAYT5D3+FQhx12wX7pr3aTajY4X99Im7VZNdv5KIlb9woV1+x69HZwJor1YfuQfPfZonF46zZgJzYA0I9N2A06fTV7uBsB+sUGJ22DndWNeA8AAIvfrHpcvV5hf3SxJh5ekOETBwiyP+x6Fge4XwIwvwG7i468zxazUnfQqq1flH9P26V05zUN9jotJr0WV5FXWDlIIt/KdxyIzjIX1sV2PSeX4yAB8sNc0Vjs6CrPeuhIvckcWdNBqwAKEDkwxFbf7o3cdFdircUHlrnNuEq8wupBEokqAx6Ivk6x3PnmuwAa+6ID4Qxz93jxjPho1Z9105F7k2zmB6zUuB7wCkeqcZB0ZyuNrkTBPAe+MUnH3WJV14uUB0nORn10BYv1jNwm9/XO9ZhbrENe4axpIz+tNERXoPVm8p+0DneOOA+UeoV9DvykejQi+v9OWg9fx98OvrPhCqT5pNubepB0sRK3WMeYu2pgd614lYOkS1Vxi3WEuasadm2CawdJF6ryFmun9a0vszVO8Dmqfot1wNxVX0b0TfD5Lo2Chi5HacbZxhXAftHTlcw427hQ2OUt1i7rW2s3omuCz7MZKTzJYXNXA2xtE3yWKm6xvhk2d9XD1jfB56nDLdZBc/fYWiVr/fRN8JVqztmGkZGRkZHRXP0HkuEIgjyJDtsAAAAASUVORK5CYII=)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p =
and Q =
)
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
----------------(1)
Now, it is given that y = 0 at x = 1
0 = tan-1 1+ C
⇒ C = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABkAAAAeCAMAAADuMkXpAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAA///b//+22///tv///9vb/9uQttv/29uQkNv//7ZmZrb/Zrbb25A6tmZmkGaQtmYAOma2OmaQkDo6AGa2kDoAZjpmZjoAOjoAADqQZgAAOgAAAABmAAA6AAAA619GugAAAAF0Uk5TAEDm2GYAAABtSURBVHja1ZFJDoAgDEU/iiOgOOCEev9jGkl0Y9kQF/p2zUubth94gWjcHeJhqkwLqJ7sCjDMnCaJiRUWgdK2MT6A3i+2AsBd4W+kVtCCdZPHKGloUzaMNtHAmZE1kZxyX5vpTD3TAOR25UHnHgE4Burlaz/uAAAAAElFTkSuQmCC)
Now, Substituting the value of C =
in (1), we get,
![](data:image/png;base64,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)
Therefore, the required general solution of the given differential equation is
![](data:image/png;base64,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)
Question 15.For each of the differential equations given in question, find a particular solution satisfying the given condition:
![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
This is equation in the form of
(where, p = -3cotx and Q = sin2x)
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ y cosec3x = 2cosecx + C
⇒ y = ![](data:image/png;base64,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)
⇒ y = -2sin2x + Csin3x--------------(1)
Now, it is given that y = 2 when x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAeCAMAAADJnMQBAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6AGa2OgAAOgA6OmaQOma2OpDbZgAAZgA6ZjoAZrb/kDoAkLbbkNv/tmYAtmZmttv/25A627Zm29u22////7Zm/9uQ/9vb//+2///bvb40iwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAX0lEQVQoU5WPSRKAIAwEx100igqK4v//adgOVnGhT11JppIAGexceQj7sRFU42cy9i6u+1yAFQTT9mzFhF1McfIfOPnqwZXstELXMnbvLpkaY0n7MUYlMSyGXFaEx3N8HU8DSRzETU0AAAAASUVORK5CYII=)
Thus, we get,
2 = -2 + C
⇒ C = 4
Now, Substituting the value of C = 4 in (1), we get,
y = -2sin2x + 4sin3x
⇒ y = 4sin3x - 2sin2x
Therefore, the required general solution of the given differential equation is
y = 4sin3x - 2sin2x.
Question 16.Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.
Answer:Let F(x,y) be the curve passing through origin and let (x,y) be a point on the curve.
We know the slope of the tangent to the curve at (x,y) is
.
According to the given conditions, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p = -1 and Q = x )
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
--------(1)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALAAAAArCAMAAADvyPLsAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkLbbkLb/kNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29uQ2////7Zm/7aQ/9uQ/9u2//+2///bK2G21gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACjElEQVRYR+2YiW7bMAyGJW9rtKvN4h3tOq9dvK1OD0+13v/dSuqw5UamhAHDQiBCgQKO8/PTL4qiIsRxcHfA/JRyw2kS7UbAH5+h39zwgUXS9uSBFbBpOKUDWKsVM+BefmOVEaJjB1xteTncMgM2zQteZXgObH69L8Xvzn7/XSqlY9yqaqvVyUNWdqgjh03zIXuIBGWh303lpZfFW3cpRg+wa4gey6YcGerpoCs7Q4Ky0K+n7VpcHJ/FGGppB8y9e/nVAsayCeIYuCfSOaEsummuWhVW8+UYu7deLpJNAGPe+MemWRVl5agceUEDt1LCgqOPYwz9SVbOUT/+/DDNK7staIsj4HRQQtkvL7whT8Fhj5WaNNZO08AihBhaXYq7eFV2amVamBZ8m87MXo629qmKTCmLFr+Lb5grzEKHBf9dXsIIiqYBh7FnCTGwRVxcUSu7NHLAtDI6YttTu+lGrP1gWp2u8akHHmqkX2psw/P9iaPC5HDn91z8Xl7ZveGWOmAl3Omc2T6G38ILm5zs0DMO55VjYOGwEs4MX84t3czhpVUngbvp/pnKYYezqAzJFgMHrL33zfftUGNm+hh0QSJzOAJOVQlS2bK6N1wOB6znwAYvuT1aE2L0EnbqblbXxi/RJkXASThK2RXMrroWj5+xxjqs/eUwDZQLTK7NVBl2SlYX6T6ArsNx/548hQhldyTBzxrV+l7JS49FpBAuRb45pE+6NupaynqJEShz6KfB8zEysrP+vaRbm3ijbo12dfZpLkauW5tfOP5fPxwmleuHGV44mP3u48o5o0EX6QOcSPFN4VDYC+r4oaA6DvpUOSxWpLHXFjZj+LiFKzsbXOxk3aXvOI4O/FsHngCzrEpX0HPUUQAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Thus, from equation (1), we get,
![](data:image/png;base64,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)
⇒ y = -(x+1) + Cex
⇒ x + y + 1 = Cex -----(2)
Now, it is given that curve passes through origin.
Thus, equation (2) becomes:
1 = C
⇒ C = 1
Substituting C = 1 in equation (2), we get,
x + y – 1 = ex
Therefore, the required general solution of the given differential equation is
x + y -1 = ex
Question 17.Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.
Answer:Let F(x,y) be the curve and let (x,y) be a point on the curve.
We know the slope of the tangent to the curve at (x,y) is
.
According to the given conditions, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p = -1 and Q = x - 5)
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
------------(1)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Thus, from equation (1), we get,
![](data:image/png;base64,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)
⇒ y = 4 – x + Cex
⇒ x + y – 4 = Cex
Now, it is given that curve passes through (0,2).
Thus, equation (2) becomes:
0 + 2 – 4 = C e0
⇒ - 2 = C
⇒ C = -2
Substituting C = -2 in equation (2), we get,
x + y – 4 =-2ex
⇒ y = 4 – x – 2ex
Therefore, the required general solution of the given differential equation is
y = 4 – x – 2ex
Question 18.The Integrating Factor of the differential equation
is
A. e–x
B. e–y
C. 1/x
D. x
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p =
and Q =2x)
Now, I.F. = ![](data:image/png;base64,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)
Question 19.The Integrating Factor of the differential equation
is
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p =
and Q =
)
Now, I.F. = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
For each of the differential equations given in question, find the general solution:
Answer:
It is given that
This is equation in the form of (where, p = 2 and Q = sinx)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
--------(1)
Let I =
Now, putting the value of I in (1), we get,
Therefore, the required general solution of the given differential equation is
Question 2.
For each of the differential equations given in question, find the general solution:
Answer:
It is given that
This is equation in the form of (where, p = 3 and Q =
)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
⇒ ye3x = ex + C
⇒ y = e-2x + Ce-3x
Therefore, the required general solution of the given differential equation is y = e-2x + Ce-3x
Question 3.
For each of the differential equations given in question, find the general solution:
Answer:
It is given that
This is equation in the form of (where, p =
and Q =
)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
Therefore, the required general solution of the given differential equation is .
Question 4.
For each of the differential equations given in question, find the general solution:
Answer:
It is given that
This is equation in the form of (where, p = secx and Q = tanx)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
⇒ y(secx + tanx) = secx + tanx – x+ C
Therefore, the required general solution of the given differential equation is
y(secx + tanx) = secx + tanx – x+ C.
Question 5.
For each of the differential equations given in question, find the general solution:
Answer:
It is given that
This is equation in the form of (where, p =
and Q =
)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
-----------(1)
Now, Let t = tanx
⇒ sec2xdx = dt
Thus, the equation (1) becomes,
⇒ tetanx = (t – 1)et + C
⇒ tetanx = (tanx – 1)etanx + C
⇒ y = (tanx -1) + C e-tanx
Therefore, the required general solution of the given differential equation is
y = (tanx -1) + C e-tanx.
Question 6.
For each of the differential equations given in question, find the general solution:
Answer:
It is given that
This is equation in the form of (where, p =
and Q =xlogx)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
Therefore, the required general solution of the given differential equation
Question 7.
For each of the differential equations given in question, find the general solution:
Answer:
It is given that
This is equation in the form of (where, p =
and Q =
)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
------------------(1)
Now,
Now, substituting the value in (1), we get,
Therefore, the required general solution of the given differential equation is
Question 8.
For each of the differential equations given in question, find the general solution:
(1 + x2)dy + 2xy dx = cot x dx (x ≠ 0)
Answer:
It is given that
This is equation in the form of (where, p =
and Q =
)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
Therefore, the required general solution of the given differential equation is
Question 9.
For each of the differential equations given in question, find the general solution:
Answer:
It is given that
This is equation in the form of (where, p =
and Q = 1 )
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
x(I.F.) =
⇒ y(xsinx) = -xcosx + sinx + C
⇒
⇒
Therefore, the required general solution of the given differential equation is
.
Question 10.
For each of the differential equations given in question, find the general solution:
Answer:
It is given that
This is equation in the form of (where, p = -1 and Q = y )
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
x(I.F.) =
=> x = - y – 1 + Cey
=> x + y + 1 = Cey
Therefore, the required general solution of the given differential equation is
x + y + 1 = Cey.
Question 11.
For each of the differential equations given in question, find the general solution:
y dx + (x – y2)dy = 0
Answer:
It is given that
This is equation in the form of (where, p =
and Q = y )
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
x(I.F.) =
Therefore, the required general solution of the given differential equation is .
Question 12.
For each of the differential equations given in question, find the general solution:
Answer:
It is given that
This is equation in the form of (where, p =
and Q = 3y )
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
x(I.F.) =
⇒ x = 3y2 + Cy
Therefore, the required general solution of the given differential equation is x = 3y2 + Cy.
Question 13.
For each of the differential equations given in question, find a particular solution satisfying the given condition:
Answer:
It is given that
This is equation in the form of (where, p = 2tanx and Q =sinx )
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
----------------(1)
Now, it is given that y = 0 at x =
⇒ 0 = 2 + C
⇒ C = -2
Now, Substituting the value of C = -2 in (1), we get,
⇒ y = cosx – 2cos2x
Therefore, the required general solution of the given differential equation is
y = cosx – 2cos2x.
Question 14.
For each of the differential equations given in question, find a particular solution satisfying the given condition:
Answer:
It is given that
This is equation in the form of (where, p =
and Q =
)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
----------------(1)
Now, it is given that y = 0 at x = 1
0 = tan-1 1+ C
⇒ C =
Now, Substituting the value of C = in (1), we get,
Therefore, the required general solution of the given differential equation is
Question 15.
For each of the differential equations given in question, find a particular solution satisfying the given condition:
Answer:
It is given that
This is equation in the form of (where, p = -3cotx and Q = sin2x)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
⇒ y cosec3x = 2cosecx + C
⇒ y =
⇒ y = -2sin2x + Csin3x--------------(1)
Now, it is given that y = 2 when x =
Thus, we get,
2 = -2 + C
⇒ C = 4
Now, Substituting the value of C = 4 in (1), we get,
y = -2sin2x + 4sin3x
⇒ y = 4sin3x - 2sin2x
Therefore, the required general solution of the given differential equation is
y = 4sin3x - 2sin2x.
Question 16.
Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.
Answer:
Let F(x,y) be the curve passing through origin and let (x,y) be a point on the curve.
We know the slope of the tangent to the curve at (x,y) is .
According to the given conditions, we get,
This is equation in the form of (where, p = -1 and Q = x )
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
--------(1)
Now,
Thus, from equation (1), we get,
⇒ y = -(x+1) + Cex
⇒ x + y + 1 = Cex -----(2)
Now, it is given that curve passes through origin.
Thus, equation (2) becomes:
1 = C
⇒ C = 1
Substituting C = 1 in equation (2), we get,
x + y – 1 = ex
Therefore, the required general solution of the given differential equation is
x + y -1 = ex
Question 17.
Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.
Answer:
Let F(x,y) be the curve and let (x,y) be a point on the curve.
We know the slope of the tangent to the curve at (x,y) is .
According to the given conditions, we get,
This is equation in the form of (where, p = -1 and Q = x - 5)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
------------(1)
Now,
Thus, from equation (1), we get,
⇒ y = 4 – x + Cex
⇒ x + y – 4 = Cex
Now, it is given that curve passes through (0,2).
Thus, equation (2) becomes:
0 + 2 – 4 = C e0
⇒ - 2 = C
⇒ C = -2
Substituting C = -2 in equation (2), we get,
x + y – 4 =-2ex
⇒ y = 4 – x – 2ex
Therefore, the required general solution of the given differential equation is
y = 4 – x – 2ex
Question 18.
The Integrating Factor of the differential equation is
A. e–x
B. e–y
C. 1/x
D. x
Answer:
It is given that
This is equation in the form of (where, p =
and Q =2x)
Now, I.F. =
Question 19.
The Integrating Factor of the differential equation is
A.
B.
C.
D.
Answer:
It is given that
This is equation in the form of (where, p =
and Q =
)
Now, I.F. =
Miscellaneous Exercise
Question 1.For each of the differential equations given below, indicate its order and degree (if defined).
![](data:image/png;base64,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)
Answer:It is given that equation is ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMoAAAAmCAMAAACyPfRjAAAAAXNSR0IArs4c6QAAAMBQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZmaQZma2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAnGYAkGZmkLaQkLbbkLb/kNv/tmYAtmY6tpA6trbbttv/tv//25A625Bm27Zm27aQ27a229u229v/2//b2////7Zm/7aQ/9uQ/9u2//+2///boqK0YAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAD7ElEQVRYR+1Y2VbbMBC1kwbiUlrcNpTS1elGo7TgpJuFbf3/XzEjybIWLFs+IQ8UPeQQIt2ZO6tGUfSw/lML/DmL4/RecK9PL6LtZHUvuACJ8vH+qLBlHM+u7sxy+cs7g3aA61eX9dl4edXXSxuSpoX613ZfqZLHcxRan4/2SvlExI9EEhTy4wYv3xeTKCJIpXzbWjEwGurFR3mCIzUrPxCIFJjQZkcgdOj2HBSg74ryfehBhwAiqcUyHrH1Ioa1ByoVFH0IMC4PJJN4sqKBgmV5kkgaAp2ODtkRRmXZvGCfW1tWC+CzDst/EVUNkoYg3TJCrTFHuEX1sIAAZ5+CjFkmPHgUkoaQ79MtDpX6dBXYBKS+CklDkCzH2Dj8TL1Iiyo71IoXOewpyhurcRNRqFqkFqHGcN3RyiGL/VA0iU9+xFrtKpNO8WWCtchCrBcy0xRSi8AyWY+9OrB1MmQb7aPiCDE6Jcv0InprvLjUNQQyIFlokv4d4rlQKuuVUb4GUKFWyzAQ8n5L0uTDECLQ+fqxDCBiThYdVEgcHxS5aD82FQOhn0oTnyxDTAjhToUFlRL64FO88EFYYrgffJG69NmjyysEYOVvXm3z3i6vLCHyymN6/lMJPqwyDFsyXUUbhNd08dGxqRwn8Qxtg3JlzfZSsV3myiLTbzhkAgte1FWLA8fL1biJUxHXRehk0JGxasJHo4s6YPzBJbpgbIkmEX5OnsnrbkPFQhJKu9FnKQhx9aKIfmHV5AouuwsuUhHVHT8lFfza6jLcK3ynrFjKGV7D9+aKLNfYm1A56qnKLRVOA+NqK+K3VwpuMgOMUxF5Wp+/kXV2J1REcE2vCFft9pgwvQKqQYvjxa/VZbhXiHAnfEIgNKXHeznp7ytiB78x1IvnvosGUhFO5CLV9UrTJYAKgPHywZAUxaFABdztKPJS4xFRJvOiCRQiEDsWdz8KrZCPunXruugn7dcKM8Cq15C0J5fo2wl4hQ84Iv86lu+35gg2itkF/+adb2CKwoKzhW7CXw2grcFK/zW6cARtIh/zWqFZ3pjt+zzm0qfdRnE212fgo+haDBjtMiby4NcKvasZSEF3DYh4XyV2qIjRwrl666NX+GuFDmfM9jKbu6PfCO0s1YbZ/jM8U9lav67KiXybTFYl3LXHvFbIlxULidvMcr9HQ8i/k6Ann+2RSFy11GwPzSlPi+a1ot8q2g6hsYXEk3DIGBIkyre5HYjzR2PfjoTKLlKIU3ZAqFVgczTehpjtDtJ+31u0ifz6O8tmzsPvQGOxJcamfCWQSPC/gad3tU1O5JtkzgjMPmNhf19FFlL1cyzWw7n7Y4Eb33+MD4unk3wAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
We can see that the highest order derivative present in the differential is
.
Thus, its order is two. It is polynomial equation in
. The highest power raised to
is 1.
Therefore, its degree is one.
Question 2.For each of the differential equations given below, indicate its order and degree (if defined).
![](data:image/png;base64,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)
Answer:It is given that equation is ![](data:image/png;base64,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)
![](data:image/png;base64,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)
We can see that the highest order derivative present in the differential is
.
Thus, its order is one. It is polynomial equation in
. The highest power raised to
is 3.
Therefore, its degree is three.
Question 3.For each of the differential equations given below, indicate its order and degree (if defined).
![](data:image/png;base64,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)
Answer:It is given that equation is ![](data:image/png;base64,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)
![](data:image/png;base64,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)
We can see that the highest order derivative present in the differential is
.
Thus, its order is four. The given differential equation is not a polynomial equation.
Therefore, its degree is not defined.
Question 4.For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
xy = a ex + b e–x + x2 : ![](data:image/png;base64,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)
Answer:It is given that xy = a ex + b e–x + x2
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, Again differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOkAAAArCAMAAACqyV9uAAAAAXNSR0IArs4c6QAAAK5QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZjoAZjo6ZjpmZjqQZmYAZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkLb/kNu2kNv/tmYAtmY6tpA6tpBmttv/tv/btv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/7aQ/9uQ/9u2//+2///bu6zGNAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAD8ElEQVRoQ+1YC1PUMBBOikBF4eQO8IUFVIqKPdCr5fL//5i7adOkaZq0ub3R0csMcMztftlvH9lNGNutf9cDJecfaNjRIdHY00MpqZgyOqTtUK1SopgyOqQd03gPVGecH5PElA4pno1Hs0ovmbimqFM6pK0QZfnBitGcI3RIW2G6XswBl+IcoUPaClFGZx8d0o7pZh4Q2SEAUPR7OqTNGA1qF8lH9nROcfbSIW2HqrjjyexHSkCVDmk7TJ2o4uvL7/oLkR2sipNvMft3gWIQNtIJWi2yU2iy7VovoJarFxGjsQWkEB/S5LZKsZE7VsH3DC93BbyKLrCA1SLDHqtXmdxiz32OvyctG6hVLiFJZm6ijBXDTJlXUTzCYLvftdJvdWltVdTeb/5M4GoDadXi2fvhM7LHdL3gckFWdRVFZiZazi/Z08JtvLtIZeMxVl7DTQ5q3Xeca3k0kLvoUU9MWVfRYorbFdbZ6rO6Hg1FxvnBqkp58vl1nWeDuThER82Y4uGI85l01hlPMJi/Polsf+iIK/bub2qxVr7dwVLsMpVSOBrkaHpRP6T4rK7LEkTk7F//I1c+GCI31xbocCVyxMHbziP4cZkeihyMcasVCZzzMjRKvpWzFR1Mc8wI3E1957FakZNZVBjs5GUFGddVA8twg8Nq00syvgjgSekGokBUed4H5ftMVUJCTJtjFUFag/UH3EzVCSqJK+MoU0z7pNxIZsEhWH0HsFD6bqvVQMwprzPM5W+Vq1V6LMulv59pvAoFeh9Odf3VMFNv9jLxeJ5i0TRHqO/A0Z6WTCUZj7wdU50wMjHGMsXgNsdut04nZi9sv5I34DpGwWXF1CdvM81hJ7nWF2+Ugzx12l7L14tXF8a4MtJQbVkDJHNElsJw1zHpNHU6HyFvMZUdvwR3QtHJQkfOHvdqg3JuBiG2n1bpnD29xROphNYulsMzQ8234HBWf0H3BOW7TKWBIgcXwQ8oS+O11Y5X93a06c44oRmpj9To30E3/bnAtrGE9vxuaApUcS1Ob9RQF5K3ZyRZ13MYBRKIKXxCv+mTpn8Lb5ttaTbQcEh7SJNnjWANTxYwrXa8luEVBFK902JG3GX6SAN3mcn2Rit0rHa9C8K18j6bXRshDd70sCT67+N/zf1Uvbqbg2LjP0j2k1BFGa72IEUHhFBRv7qbg2LMBnRIMbuHdfSru3xAGdXX3ah0SGGrIyTMt2hjUPyjSBGbj1DpvLrrQXGEpi1ChxSx+QgV0z5jUByh6WO6GVLE5iNUjFd3Y1AcodgToUOK2X2ETvvqbg6KI/T6InRIUdsHldSr+6U5KAa1XAJ0SFHb75T+Ow/8Bp4pgYYe7SzbAAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
Now, Substituting the values of
’ and
in the given differential equations, we get,
LHS = ![](data:image/png;base64,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)
= x(aex +be-x + 2) + 2(aex - be-x + 2) –x(aex +be-x + x2) + x2 – 2
= (axex +bxe-x + 2x) + 2(aex - be-x + 2) –x(aex +be-x + x2) + x2 – 2
= 2aex -2be-x +x2 + 6x -2
≠ 0
⇒ LHS ≠ RHS.
Therefore, the given function is not the solution of the corresponding differential equation.
Question 5.For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
y = ex (a cos x + b sin x) : ![](data:image/png;base64,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)
Answer:It is given that y = ex(acosx + bsinx) = aexcosx + bexsinx
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, again differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAaUAAAAXCAMAAACceaBWAAAAAXNSR0IArs4c6QAAALRQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpC2OpC8OpDbZgAAZjoAZjo6ZjpmZmYAZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6nGYAkGYAkGY6kLbbkLb/kNv/tmYAtmY6tmZmtpA6tpBmttvbttv/tv/btv//25A625Bm27Zm27aQ2/+22//b2////7Zm/7aQ/9uQ/9u2//+2///b+QubKAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAELklEQVRoQ+1Za1cTMRBNioUVQYoWn7Ao2uJ7C9q6kP//v5y8NpPHbNIuHjnY/dD2ZJPJvXMnM0nK2PbZemDrga0H/nMPiC+HP0pd0Bx9w11vp5zvLksH33W/hnN+RhsdwOuukQ63J+rjnJ8bvmN1bJ9gv9xOnyMAV9Vo1lbDVSu301a0SooXQt7vKcvLj7tyJD3Wy43QYSdq7GhisqZTibWPZ66TrxJb7S6bSU7ygrgK7ayoJdOjkuGFkGdk0rwyjEgkfcbXcAtFaOUE8GYSNQpTzLVBiyXgxJpHbwtEyHcJ7JSp5CFmhlexSkzzyjDqV8lH4GiWu4VQSdR7aaeRKuHFFHJa7A/PdxJOqR2PlIfY8ipXSfMaxIhSqZQOY4RKtllc7XM+QYIFKn2/4CO9TnCKNJzaE/3y90dRj+X2Ahr4CNai/D6UDTfn0DBZzuVmA7IvUU1CO2bUSg6QH4tKQkRGaJXsm2bHIU+EI6KteRGMskiMazxipW5557xCqLQaqXQMobcUc/1bP75KI9jbNca5c7f6NKe2OmXXUMcX1Z6Yyz2fbLiZnunvGlKqqHeX4j28kVNQERfZ6UYp7G11MNMYnBFaJcOLNRh5JJNHW/FKM8ojSahU7paAEISheawglo2chebcyO63Uy3P3GU1zUk2eJnT9lC8pVXVrz2U3WAtUduV0E43yqikpILBzkiJSgh5YjEh2gp1mlEeSUKlcrcQhBxcnLc7zpGYupdxfqiSI2bM2tyuv9XnJT/4qt621dOJ6RbOEtnpRjmVbKgbIznEjPnIoyDFwelU2gCJzMP+AljHLZ1X+jMeE9cvKq9eJHYPpEoKntsr+irpVQZlb6zOXGpZJh95VPHsdKNClTojBWsJx1c8LaaNVVobSbyWIjp9brFeUYTIjCcTdF/G89ZSUJfCbVG3T0JrCUhcVyrvv3xFbP2j7ZUirkaFKnVGhqrk0XZ1KWKUR5JSKcjsiRRjCaYJuaAyPFV5ytYlNS2moH5Hm3nbINOttipLki5f5zNb3qLAjux0owKVnJGCPZ6uqEQpxLRRTG2AJFZpDbcQhJyHjC1ZlW9e4/sWP+Nx2AJ+1rlKnSuMqzWzFT9lYoHOs41s+DSDF2BVSqU2eyd7TMyhv2xNPqEdOwpGQB5QHxInMlJwXkLI40kxbXxeWh9JrFK5WyhCCK85o1/CaennFB1kfJWOLzgf64KizuieSrAF56M3+GYIGsYfoO8V1Dp5ZXRzDj+OlqLmI1hLnJIpsGNGgTRQMdWHHPsMGaFV6u4eEPJEaCDa3t3D2kgSKpW65RdFCOEtusdD/fvu8dJL5C+25u/xiic3vMjkWGxoUEeSUMGdOBap5058EL6NBveoBOUye9ef4HVfVWID/od5eP8v3VuVNorihzron8Zd5m/Nh+rzLa+tB/6OB/4An3XhmqlRFssAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, Substituting the values of
’ and
in the given differential equations, we get,
LHS = ![](data:image/png;base64,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)
=2ex(bcosx – asinx) -2ex[(a + b)cosx + (b –a ) sinx] + 2ex(acosx + bsinx)
=ex[(2bcosx – 2asinx) - (2acosx + 2bcosx) - (2bsinx – 2asinx) + (2acosx + 2bsinx)]
= ex[(2b – 2a – 2b + 2a)cosx] + ex[(-2a – 2b + 2a + 2bsinx]
= 0 = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question 6.For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
y = x sin 3x : ![](data:image/png;base64,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)
Answer:It is given that y = xsin3x
Now, differentiating both sides w.r.t. x , we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, again differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAP0AAAAtCAMAAABS1Z3hAAAAAXNSR0IArs4c6QAAAJ9QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmtpBmttvbttv/tv//25A625Bm27Zm27aQ29uQ29u22//b2////7Zm/9uQ/9u2//+2///bt9STBwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEdUlEQVRoQ+1Z61bbMAy2OyDbuJStY4yGS8NuGWyjaf3+zzbJtzipZcdJzTk7kB9tIZI+fbItyxZjr89rBBj7c8H5/KUGYvvpjj3MVi+P/przpWTdvE1hb9WeKWK58Naaff0xiYhRS1KaIJwJrynk2D8kLnutNoFPmmomPGW2TiTPMnlDhiQDXgOp/hTZr4H8eslEyfnhU1PwcAa0amnDN1o6D15TXDFxA+t+u+DwQBREefgEcYiQN2qj6aQpWjfT1GLSlaSqs54Srt/8hI+joKZHLYY06X0evO0C03x3QeFf4jq49/nUJrGLKGfC85kV5RFb45Sgn0zekICZ8LxmYepXqvyhnkzePDcejnN/3UMCPL+EpR94vGoZp34uvHp2xzafO1mPsYrHij6vWkb6mfDEPZ/Nfxdd+mvM+uHB96llZO91MxPeOrzdUaji20ksao5qffZjovtpeIOgYa+LbHck+fJDcKPo6zXvw6k1FhuRiOfYI6FFOb8ZNfSi7CSL3oFMPLzj/GDVYUQfqH3S/WD08OhYpUBDnX+WNIQGtpcseuwrzK5lr3yunapClM5M8Er3+MWTk1aIQscm2YD3alMinwrf1r29xR38LnufdNd2BM8RjkIPYBcTMdXy5prDLvIkL2Lw41fR3hZWsJdUeISs1W2SO3k77CUYStOPxnPM4TFwBmbx+wQzqnbFGqOgY9QGvNeHQjwhihuY0dK7pjhe2RHf3MosX8H0N1TlsKinz15Lk8jmEGrN4TFws1gCJnyXEDnrCsQhCB2YsXjulU/kxlN7I2vg5sSylyHAdIgnaJlQ0ClzhSaPbj72VjrK3poztmREMfbWFRLactv9QeF6VFBUnovhuefH3/Fbjz2wV4cCeB4LKdMUp+YWSXsMU3I3xFqaRfCMOYOivuWncSUETY970ht7HQJb2wGQ9LCHPCDjUNtpRI49imlpwov2+kWZ67JXOVG74hjzQtM8vcPiE3cugx4LgPax12vg8ovJZ4F1b1YM5ZrF2ypznrGXY64zSwg6aZT9wjoH45JnOKRd9iq7yxoAKsntQvlk1wT8drOeI016ZvYYY87sgOpiDt9qV/5GoffAXqPLjHsB5CRT+YFhF1jobNAxUUEc9KSm9vtWmnZM47Xmag43lF9X0rZE0q7EoPtNkgEdPU9fRdVem+tC5na4FOZL+YEZF/y5hd9zIF/C5qH+BQmAqvWMtCVP4bnmoLI4uMPVrpCsKzHoblU6qKO321cZXHdbSimNs33gORPJhd5tE0Qd83QWUs9cSWe8PeC55N3j5a7paEfP11dJO2+nne+n4znkW2jTJHGL4EhHz6uyh8RJmciH1zZJnPo73NHzquQkb/tG7hFhL4Btk6Stv3VHj7LvU9mLL4SRfHjuxbypv21Hz++NTyUn+Yx4nbZEWwSH2IxQmRScjHiuaV0wx1wdoRIzGXyfEc9pkjj1d9CbESqT2OfEs00St/4OeztCZRL9jHimSXLl1t9hZ0eoTGL/3HiTnH1V/j8i8A8R96QkR46AqwAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, substituting the value of
in the LHS of the given differential equation, we get,
![](data:image/png;base64,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)
= (6.cos3x – 9xsin3x) + 9xsin3x – 6cos3x
= 0 = RHS
Therefore, the given function is the solution of the corresponding differential equation.
Question 7.For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
x2 = 2y2 log y : ![](data:image/png;base64,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)
Answer:It is given that x2 = 2y2 log y
Now, differentiating both sides w.r.t. x, we get,
2x = 2.![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFcAAAAhCAMAAACbbceCAAAAAXNSR0IArs4c6QAAAKJQTFRFAAAAAAAAAA0AAAA6AABYAABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6ISAxOjqQOmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjpmZpDbZrbbZrb/kDoAkDo6kGYAkLbbkLb/kNv/tmYAtmY6tpA6tpBmttvbttv/tv/btv//25A625Bm27Zm29u22////7Zm/7aQ/9uQ/9u2//+2///bJnHHewAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACBklEQVRIS9VV2VbCMBBNRBRbF0BBFOJC3UNtKM3//5ozk6ZNWmjlAA/kgQPhzs3MnY2x4z2SB4dxPjoQrzwA72rC+QF00CJI9PP+/U0v54w16vALAQ23zusaXv3R/yl5svsXFnfg8ZYjB18uIhsPk5W4SMo7Le6cX3SPj0veQp5ez1xiFfLBO38srrQY1TyTeKXanKbQNx7VdUQwqJjkbeVlslcN1FWhVhvSZK2dt8nhNESVtOC8l6QhaqqAVsGl4U2hOvqYIf0WQuHzkxIIVnUJrce5V1pgTPgjG6O55U3DKVsJlCrqztmCz0ogMDTMAxutRFuvrukfssSYoJ8YvBkwFxjlAqMnzkGfCWZtn9wEG+cxUvzMeUfmkRxoedcUhfUX7ZSX35KXOCN4JgZ9XOA/eNHxyC90z19KbWdqIrTABn1NPVCwtw9eJaO/JkuEoV7xgUal9YdipBNxH6WoKOByheROTRXAxoYr+q3SeArChjTGULVDbCtJCcevBdDrt+pyKxxRjUM5m6BeS1TEAivuVsXGeQaFY2tng16mHrPxTQGszLP6UIf5+y2GLTsEhYZePp1YoDd/8+UWh515GjrVCkU02DybKIT4CuQdfK4FFssNOkBSNvZzyiUkz8rJvjt3ybu4apjJWz9ULLflqxbn3t7bmsszyJfbIgx0BIN8N7KjtP4DzkQ9vovDbA4AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, substituting the value of
in the LHS of the given differential equation, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= xy –xy
= 0
Therefore, the given function is the solution of the corresponding differential equation.
Question 8.Form the differential equation representing the family of curves given by (x – a)2 + 2y2 = a2, where a is an arbitrary constant.
Answer:It is given that (x – a)2 + 2y2 = a2
⇒ x2 + a2 – 2ax + 2y2 = a2
⇒ 2y2 = 2ax – x2 ---------(1)
Now, differentiating both sides w.r.t. x, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGwAAAAuCAMAAAD6KAyYAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOmY6OmaQOma2OpCQOpDbZgAAZjoAZjo6ZjpmZmYAZmY6ZrbbZrb/kDoAkDo6kGYAkLbbkNv/tmYAtmY6tv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/9uQ/9u2//+2///bKxni8QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACH0lEQVRYR+2X21LCMBCGkypEEamgEJBgFW0Lef8HdDc9kJYmWwpUZzQXnaFl8+8hu/3K2F9dCeeL/mJP+hRLRY+R9SaWPnE+EgstOR/EqeDB2lU+/XHP+djxVKF5RBQ/FXOmV1AzLQcxY4lHSw5jrZzP8YmW/nIoI4EHJLp5g8vQfyzdCUdno4nXeh/ic7MFXvTSmcRsm1IM0pavItZUjFw5zl2wxLQcsgTjdNZsOxW+okTuEmRbWmKYR+XJOTgTHyI79mg/e8ZCeBaGk9cMhB9nnn+bs+Op2XK9D4mKR8Er202zCaK4r8CpmLDdi6v9tQLbxLsBY3rDg/GnMGqJPw0b6LKvsHmyQZsGEBkn1KwcJ0QWLjas4dCT5/5yYnK86iswmDT8wdNj3YN67/GF4uvM7hE4LcnBco4mNkB1UWMMGrw+aDs7oGXev3Un4Ldr04a/Nt06Ni/FOrvrNvzRNF71gNRi7g2mULfe1L+biPUW+O+WwBXXETo1r4rP2S4kIOBiYvh+iDp8IOREjCODRlrbW4DNFhhdia8kYtYGaW1TBWkkMbqaywMRt0Ba2zSrcyuMLsxsbqSR1hLT0pBYK4xuEmMnDJaMNwHOKIy2k2FHRiPtwVIBHZtFYHSlZhYRA1yRSFvYRubbBxNJYHT1gJRE3Appc9v0DqaHMaAwuipWEPH8FKTNX+NGjMBo1xzpdv8fo7vlrdLZV8Lotp59A9WBLR0E7tvXAAAAAElFTkSuQmCC)
- ---------(2)
So, equation (1), we get,
2ax = 2y2 + x2
On substituting this value in equation (3), we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the differential equation of the family of curves is given as
.
Question 9.Prove that x2 – y2 = c (x2 + y2)2 is the general solution of differential equation (x3–3xy2) dx = (y3–3x2y)dy, where c is a parameter.
Answer:It is given that (x3–3xy2) dx = (y3–3x2y)dy
- --------(1)
Now, let us take y = vx
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, substituting the values of y and
in equation (1), we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
On integrating both sides we get,
--------(2)
Now, ![](data:image/png;base64,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)
---------(3)
Let ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHkAAAAhCAMAAADK+fThAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OjoAOjo6OjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZpCQZpDbZrbbZrb/kDoAkDo6kGYAkGY6kGaQkNv/tmYAtpA6tpBmttv/tv//25A627aQ29uQ29vb2////7Zm/9uQ/9u2//+2///brMbckwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB+UlEQVRYR+WW2XaDIBRFJc1A2ySNnRM6mSa0VSP//3dlVkEBY/WlPOES2Jw7caPo/w4MFqOIb+Ak45AjxsE1Vv1rOP2MU6o83QEwhrUlZ09ps5SJI2iRktfhra05WnN++WHavqu1c7jzb9EcF5l8Lo/+o/QK8nLFyHh1cG2yyUW8Tk9ozi0vBkE3lS//FfAWcc35tUu65iSz9EccmkGw2oPnCngj51/31BHeka2JIEdcVutQnBxOSlhtdXYhTE1jUc6c8OLhSND2nVsJi6A9c9AYFAJWBxxCxjRLVJ64RfsuVAnUdjJBAMzTHAqrKGvTiXKUj9L0P5toZzk0E8QMKx3zDZVxelXhMHKE2QWxKZHGLVeTcBewUeqoy9QLxIRHSencytQ6KoebqLjV5pHHKrJtzAaSuShQMzN3ZtXcdnKA4wPJzMeJVTmUn/3WbrhJUGzTfUX89GhmbxH3iW2Vz8zlbeEho8jC9MtnlSrkDdKomLYUOspWta5it441zGyOzi8HPslWG2amf9e3Sol2v1U8zY1ksJqwju+zJHveZ14rKmTZHCW01mQABLQWAXnZvKRO0k0YTwerFp5Nadio2jBF0i0KjUr60P4lyjjLJOnvIt7hIW0t+hXmZ0kqmzA8XfbpJ7zWski6CQvqXL3nOxaMR+pzyyH3/gJDrDHeyRRxkQAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Now, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Let v2 = p
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, substituting the values of I1 and I2 in equation (3), we get,
![](data:image/png;base64,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)
Thus, equation (2), becomes,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWIAAAAyCAMAAABReyELAAAAAXNSR0IArs4c6QAAALFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjqQOmY6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZmYAZmZmZpCQZpC2ZpDbZrbbZrb/kDoAkDo6nGYAkGY6kGaQkLbbkNv/tmYAtmY6tpA6tpBmttv/tv//25A625Bm27Zm27aQ29uQ29u22//b2////7Zm/7aQ/9uQ/9u2//+2///bYqjixQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFS0lEQVR4Xu2a61bcIBCAE7WX1FpXV2uvNtZqbDVd226aDe//YGUgFxJuAyfZxZ7ww6NHYGa+DMMwEEVzmwnMBLZPgKTP1w5Sf5/H8QL6O45zEPHfdXVDVZ3dRKu92xmxix+4IYaZy1ehIX58CxptvZHVYRw/s4tuEaP1zE/DChSb83j/Yet8qcBs7ybapGxNG1uNWNKTpF/U41YsFIcTi8vj+3xHiF9SDnms4dTR44hlPXWIc044HMRg5m4QMw6ZXXYbKAZ6CohJGscv1mUC67GghAv4bu4x3Lae/P+/O8Sb6+N7q94IxDXOghKuljFtM2KOFWgc21NeDOIoh5jOt7mmzV7MSDwmfoEiA1dlrd4uy+Q0qs56W+eMmDtbEfc8TxU1UF4MnQrYP2cvHjIE77M0FOKIxuGsn5zMXkxSgFtgkzbaV59RsMD+8V0/rgeF2J7+2zzN4/8ETh0bRImn8+K+noO8OBuGnGAQk+uEbhoHnz0gGYeUie1MsQHJC2xGIes5QAwZW7CxGEWX/DhyOWaTq0OKOEekvVbpvt7oO86qkKEDlWkenjcpkNyNpCd2fxOG5ZesglC+tvkyH2OQ7H1KMyA2idMjQoyqlv2sRp6s0EVlvjdBw5W7ikW9jHld0d60kv0PwiYvNogz6Kof9UjvAfboMco+r7ZHE+iQZbnq4oGkl9/A73PclUWgiPvxXqtkHp+syXfqn3ZrdXOQlPs/tiyXs4MXg4t04yeNuEwAD7m6jQapuWJFcENL6vVHUJ4hd5BtUFRCeqCvGYnlLljePAp3IcYYLXSS2Qy4hTCc3x4oJEO/grHUP9oNROXFMh5WIIFWDVJzHeIyuaT5KqRA2f5t9BPkCT5mKMtxm+qY8qupPmS2DYDpwSQoJE+JWCEuo2qIWBWI5VGEwUI2ZihjAn7LwgPbI3GIVeUuOh3KB3WSR0EsFY24QQpDwUvEep0CsTyqWvZNHIhrK1Zx40vVEpIH+Fkjpn8Krtv9KiuuKHd1iIfdcZIbxF1v5G/mAAOIVYZS13qzqP1R/WEUo4aIrRGRz8HwwrJZ4QMFi5qDcpeLF6skj+LFss0d4r6h1Jt6eavsxQolnQNF95koMprtXTaRkitqvCKRy131crQGqr5bdZJHQWzzR0FcdfFeDKw6xAxSiwdcEdvA0Po2EkoMbVRCZRQsvAzLXfVytCqgkzwKYrUXqwyliVfvgCYjVo3iRW9I2uyNlRZhAKuAdflWkxeDFxu/mFTuQufFGslTIVaLIxllJd4SDBBr8dC1zo4e1lbwe5pVUwHjJwgohtWZmLUsJ5W7EOcdts9qJTvnxZg3bRpxf1JqP9wottcEPcR6JdkB4uCToYqj8czqHFLwvxAykAcI+SNaDnd2yQJiVHnE502bYKjVDbsOLqPKRO3ifGdrtlC3SlutiaXShpHcnO6Q5REQ7PqmTTQUz9hhVLX8oEbMghK54zusY724Tj/M9WKkZM1rID0M1zdtPUPRjB1GZae6Ggy864vHqKpr1MZJ1l2PCpP2yyPub9r8DEWPKo7W9ion+tO6dERKRiAWXgPRnGfBlfAtH7mYgOoL7zt2gxgrGYNYKI8E96aNpQoUcY67AEJ9NVwntGQVYv1roPDetNFcj7etI0ZLRnlx4K+BdhQo2JEDcSJCIQ78NRDKUNzqd+01HuKwXwNZ6g6u2Fz6myseg8zAsTwSTEYBq7V9QupCZ4y+KMkclXt5JCTEY8CacA5fVL7jJjQl1Kl9UfmOC5XDhHr5ovIdN6EpoU7ti8p3XKgcJtTLF5XvuAlNmaeeCTwNAv8AO8fwiCq8ftkAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
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⇒ (x2 – y2)2 = C’4(x2 + y2 )4
⇒ (x2 – y2) = C’2(x2 + y2 )
⇒ (x2 – y2) = C(x2 + y2 ), where C = C’2
Therefore, the result is proved.
Question 10.Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.
Answer:We know that the equation of a circle in the first quadrant with centre (a, a) and radius a which touches the coordinate axes is :
(x -a)2 + (y –a)2 = a2 -----------(1)
![](data:image/png;base64,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)
Now differentiating above equation w.r.t. x, we get,
2(x-a) + 2(y-a)
= 0
⇒ (x – a) + (y – a)y’ = 0
⇒ x – a +yy’ – ay’ = 0
⇒ x + yy’ –a(1+y’) = 0
⇒ a = ![](data:image/png;base64,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)
Now, substituting the value of a in equation (1), we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ (x - y)2.y’2 + (x – y)2 = (x + yy’)2
⇒ (x – y)2[1 + (y’)2] = (x + yy’)2
Therefore, the required differential equation of the family of circles is
(x – y)2[1 + (y’)2] = (x + yy’)2
Question 11.Find the general solution of the differential equation ![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
On integrating, we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAXCAMAAAAiJm1nAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOjoAOjpmOjqQOmY6OmZmOmaQOma2OpCQOpC2OpDbZgAAZjoAZjpmZmYAZpC2ZrbbZrb/kDoAkDo6kGYAkNv/tmYAtmY6tpA6ttvbtv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bSD0ZwAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB00lEQVRIS+2WXVfCMAyGk6lIxQ/UgcKAoQ7Zxvr//55JN7bWUU6L5RwvyMW4WHnz9E3SDuASZ3ZArofBMoTUaqA2j5NgfCG1Os+yYHwAp2sVOLeU0V8zgNZ2ghg95R2SqVnFqIKZ/8rnpCUTw54MX3L5Ea1cOt+fz65q1TL5SsE9JRfEt1uQkeO8YK/4sRE4NvRleqtslgniIC8F2nflrAUpa2V1fVSYfFmbQyaDXC4HOZSC1pZitILM6EQSQXytJWgZFHY8Dy1ISUZnMvhkcvXVcFcx5S7vWz4FqXD6kfG/2gqRBU3siX20eC+Zlsfgq2J2oo41jj75t/GP+FSaQ8FLVE/YwkOL8j3s+6i3U50Pvu/whnxx4JPJEIpuZwco3bWoEEaj2OrLSbY8LQ58QAVO247u19dLq5q+tU3Wmw/uzjq49SDt5uNYfanyz9N94x5wz0eL+qSKtYvJnN9CTST3UilmsOMbVp3P6mGdD9pIPcmW8NCSKQnVFHWYfDKJZqDO591CINI9QucaztWDj3sbRaGXpEfprkVnaUT+aYl+3R9yTSjX79r9dsSX7lUR8FPBKaHHIuqGo4eLh9RZlspkvPzH9vH9q3/vnMWDi2hgB34A2WY2Q2EVDb0AAAAASUVORK5CYII=)
⇒ sin-1x + sin-1y = C
Question 12.Show that the general solution of the differential equation
is given by (x + y + 1) = A (1 – x – y – 2xy), where A is parameter.
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPAAAAAuCAMAAAAhv2T/AAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOgA6OjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///blZ0U4QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADEklEQVRoQ+1Za1fUMBBNVqEKq0VQqIAFCyrtNv//7znpYzeZJGVy2kmPsvnQczg75ObOdB7pFeK4jh4Ie6CW8obdP0lAqCzqBIRFEhAi4ybjj7BIAnIk7Hqg+SLlWXajCilPXppMbn4QvRRjNoCIUoNUKUpG8HRNdi3ULeSwKk5ehKh5+I4gooT9VZEggYKEy46mLlrVu5/wOI0JHNX2AKLdWn2m/h+DXZtr9K6e6If6zvFCGyCAcrZl4EHe0jiLKk5FreO9+DIJi4olachnNs8C73TJkl0mSHv1VafOakuHdchh0eafrljOYoBAzrQ5S52gurDa3IvdZT9plZKpnOxBVAkINRcMibN6kJvt76xjXHO9bCPIdQFtvs3lqowNt9Srvmuk+CxnBMnF05OWO+KyO6lie/sPBFjdSfnxcQnqMEufx/XgNSZEPajtCk+peWLpp7Zj1yDcjfq+i3WKu+cahEsdXG8jTzDArUC4v9G1OTx1c7MX+8jKThguz8MaufSE+ydeakxt7Ii5f3dA+CxzN0X/H6rCJMJLlHDvHuwR9kWxf6Whf76NV1oMRcsz8v+fRUtUeuZ/S21JF2hvzfIPHn/gA+Vy31cCObwsCM7jHVAgT4Ttxb14ntGtfl0SPnglAYkoxM0H49Cv1Fn7Z/At9bocAYKOEAFCJG19Io0h3Jw/dt9xKSsCxD5CDEi4V5uKwrOVwiaaR3nA/pggbCkKYRBXeHB8TvZq0POmolDZJctCc5WHCMKmojAF4ggPDIQNRaEGvrVxcbTRHOUhhvBBUZgEcYQHDsJ7RaGfxgbC7mRuKQ/uz4MWE3qTRkXhVZCD8ODD6MMzb4UVBdu9jl1MhMEdgYaHdkFmHBHWTvMrCvgsyC6KcFBRsHfBZiyEg4oCQsN2MYTDioJdGbHwwEI4qChgNKQ8OITDU9qEomD1Pkd4cAnPGAX3qR9SFDAasrNjc5dBxXv/LXA1DisKxi7Q67HwgMrIFEhEJaMqClS7COj0plRFgWqXnkEkIlVRoNpFwqc3pyoKVLv0DI6I2gN/ATfHWihHajllAAAAAElFTkSuQmCC)
On integrating both sides, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYgAAABOCAMAAAANW3EJAAAAAXNSR0IArs4c6QAAALpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZjqQZmYAZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kGZmkLbbkNv/tmYAtmY6tpA6trbbttv/tv//25A625Bm27Zm27aQ27a229uQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///b+nPSdgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAHcUlEQVR4Xu1di3bUNhC1F5q4pS1JyVKgD7JASpyWukkLsbf2//9WJT9kvSzdeOVY07M+HM4JkaU7c6WZkWYskuT4HDVw1MBKGijT9HKloecNSw4wKmZJjIgkKsDNTZpeoKp2t6uyx1gR5ACDys0vEvYnyPM4RJADjOm2+uZPrKGnVfUqTZ9nl80uTU/uqyzdXAfp1uyEHGBQD/nJPdjS2azK3ibNB+Yjmh3vsFyMh4QcYEy9zS6MWWrV0/q+4glbYsUpNvzDW5EDDIpYZUGIqLe8m9ZH8L+ad0sZJtY9McAgEYECOImIZnealGHsnU0GcoBBIoowob9EBLdN+XJhLDnAKBFhvCpfBr2PSOrtD2/CRGI2GQpqgEEi8jByJcXmY7J/3S2vPNQG0SYDOcAYEc2OxzgBHrbd3Zz/k7VMlIH6tMEiBxjUbTC55PHKxWLXhG1TFmB5ScAgEfU2sFwscF0wdk2YAyIGGCYicKDZ7M4/LLggGBHEAN+xo5/Nmff0YgG5Uv+o4CxhzZqdFgjHDlgXrUhf3De/mxHRX5pcVRZ4guE6hloaRMQOWJOqyrh5sBhr/aA6drmoEzG969F+U6ZLGnRo0jsbGUTEDliVRorx6m2qPYq9il0uWkTkQtW9kl0eTQ3EFSJ0ytb7uZ1Xhlj8H2MGbK7tmUQcbkmC90BrRRji46apWPJYKAAtBhFRAzbX8PTJmOasY5PrM9v+nEsERk+EDlibfGU70YHwNTIi6pcfk1s5moidCAOwRkSz27BcPrChC5RmCWCERBfV11Ka1UKEtiUNOfS8vhTAehfNTZamT3/xHnHkYRJ0kASVbHPkN6pf5Z8KZ06aHGBINSxEDJQXAoarvpvMfNxIur+doqsbghxgQDMh5fL4LDZUvVVsSiFtUuTfFW4eghHxaIAxIkKlWXw+i09l5SxFiyNETq9kPJQOL0AOMMZDX5YHNvY0c/qsRCuUrH9S3VdPU3cg4yQi3GnxowAGdVtvw535OZ2sXp+nJyfdShmlIQcYJmJm2dzda72Uz+1k9VpYXvjU3D5L06+6frpyHP/T1U/NeNYCDEKdWUe/f5XqmWOPk1U9RNK8ZxFUzitwdn3YBlZ+kAMMEmETf//bJ583OPvUVhon5flg6T1O1shHfc9ebMkZdpTgVCcHGCSiMF1f9S1SP9wRkRT93sDnZPvmAtXoT/JhaWHV9uQAY0QYBwhGuD/Vz6BZi2Isr3TfTLTPv1/43zc92furs2H5QaWU5AAjPNQvry1F25ox9xGBfa0g7M7+qk3N1j+264mvI1Hy4S+3JwcYYaFTg8UwySdujo6ErYGcrCil//ylNUVj8HqXDabJH5iSAwwSYW0GLojusyD+QEtCMjvtxC8uxeDdgX3b0awSPnKAMXbg6HD0vroftrqIUcdc3/L5xmiRICeh904OMMbDONF97Uf1A6pQUujMTFU8eO3X0vgB0KzUCDANOlFiAexTbP97LIbkconjc2ADoJh/FkHdttaIJa+u2YZO+ClhpECsbTNygDHh/A6zM+dXLOuUPu3SOVJoOjWIGhAV6bPLtuWedyO2hLO+UiQHGOPBoQtLAD/0KbZk8iBKe3WuTxXWd2WiD3smY14H3mRNwKB4019sOgSzOlmNiG4J9M/EN9hz6rwnAbuIWBPww4iwXONwEBGYGz6ECFZQdHLPkn6C8MOIWA4wSMQwV8xrHFxE2BI5Svvl5BKTm6ey5TGdRKwI+IFEmNc4OASzmofHJoJPHTkz5SJiTcAPJUK5xkEvJBzT/m23hlxG4eG4IuxlzR26OaZpNPdV9nwoOzDGjwgwSITQqXGNQ+Q+gsmnumDiPkLa5mrXOERPRP3mZ/msijgR43GFfo2DIljDv5Tsk81sc2s7rHPsI9jsVTvol+sc0yQAs+Mrbf8uR8zxAAZNk7Q/0q5xUBSbp2+Tvfje2XrK4CZC7aAHd8iGruGX4cn7RnVFxAMYJEKqpdDyDCoRcrLZXn+htDf2v0q2esDmzwyZUvSDs43Phq0I6ZJIjYhoAINESGdo3lsRBn8C6M9+IqRHW8se+o3R3bqAQSa68iPoGofBNQC399nPBXXfgm37NEGA0Yc34gAMEtGdaSPXOAg3iRxE2/y5kceYVecNHMILF9S773UBg0QkbVUGM7m+axxEhlQr8raPYzlmM1KswHG6rXOsjERK6a4NGGQCgilXSEKKMA8VTBcPphZ0OcgBBonoloT3yU/7VpgeTAMiOhBDASlXKyxygL3K7RsgZRGt8CVzJ1ANB+tYt8uigxHVrNoB/jo5wCATzbsxeznxSltDz/dQrC3Wq6Zl0YH0NuJDrYORA4ypjLX623fLX3+8eZHs/wA71SyY6GB8HQ9/zCHJAQa1tkQz73yHKgaXQDbRJznAoG58XwShH6qAwx3ejBxgUGSfon1yg8OEa0YOMCq65399QEIfdKgw7cgBRsV2qjo2D+ENi2MEDDLh2vq58mlg9+GbkQOMqgC8AgLtbvl25ACjKgEvRUG7W74dOcDLq+Q4wlEDRw38vzTwH7p+QxMfiz23AAAAAElFTkSuQmCC)
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![](data:image/png;base64,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)
Let ![](data:image/png;base64,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)
Then,
![](data:image/png;base64,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)
Now, let A =
, then, we have,
![](data:image/png;base64,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)
Question 13.Find the equation of the curve passing through the point
whose differential equation is sin x cos y dx + cos x sin y dy = 0.
Answer:It is given that sin x cos y dx + cos x sin y dy = 0
![](data:image/png;base64,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)
⇒ tanxdx + tanydy = 0
So, on integrating both sides, we get,
log(secx) + log(secy) = logC
⇒ log(secx.secy) = log C
⇒ secx.secy = C
The curve passes through point ![](data:image/png;base64,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)
Thus, 1×
= C
⇒ C = ![](data:image/png;base64,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)
On substituting C =
in equation (1), we get,
secx.secy = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABYAAAAZCAMAAAACPC7cAAAAAXNSR0IArs4c6QAAAG9QTFRFAAAA///b//+22///tv///9vb/9u229v//9uQ29u2kNv//7Zm27aQ27ZmZrb/Zrbb25A6OpDbtmY6tmYAkGY6kDpmkDo6ZjqQAGa2kDoAOjqQZjoAOjoAADqQADpmZgBmZgAAOgA6OgAAAABmAAAAciJ0mwAAAAF0Uk5TAEDm2GYAAAChSURBVHjavVHZEsIgEAtQW2wVi+uBR61a/v8bHaAczvRV94FhkpDNLsBfqre5MspO7ZKYn8US3JlFa1IA29ysfZRm7FgBZHeon1OVYXkVABmfSBXxdGoS4FXjzn1yJGdSH6x7ycdoKN9OvG48m+Kxi460BnoV0cxPVZqcjCha+3guzyCATs/yYRtu8tUCjHRc6T1YU1jrDPOveYsRzS/+9QMF5wpBPDGkLAAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the required equation of the curve is ![](data:image/png;base64,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)
Question 14.Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.
Answer:It is given that (1 + e2x) dy + (1 + y2) ex dx = 0
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALoAAAAvCAMAAABjeyAzAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZjoAZjo6ZjpmZmYAZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkLb/kNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/7aQ/9uQ/9u2//+2///bj0p6XQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADJklEQVRoQ+2ZXXeUMBCGE9QtaruKtmqL/cIKW93Ikv//45xJgA2QDRMIp9tzNhecXiRvHmYnk+EtY6dx9BEQnN/Mh9zE0WMZr7ZKKYwkAUqEQGditc3XmhzZA0SDgF7GQfbJ3/5oNwskOQofaJ/iQ50usGEgSSd6+ZXz8/hGppyvtmXMo8fRN+1NAIUIw/3vQabvnhBbS7IMJfMgB8nGVMbXTN5CYsoUQyYmkIPCM4AW8ZnMgBUCXkuyDNRkGiQZLeyZAsYzlb/5DY8z35gzVJCpsW4vieHIP3sr0hZUCSqrxMSH/OmdLlpB4ephSILm+ZoG4j/L2AcjB/XNV6NKOA78yYboLPdPQCqAGSLImMw/L7WCMUzJ6urb/p2oTMR5Okn1/VEln67a2BHXs26a4ypDEvKvSvxPD3HvPLpnu0t99WW8F0CShuBQUIr9ZQRJUkvKDPTEJFHKzvIXj9Z/YsUupv24BVwG340z0khep3BJ4FGYEhAKuzFHLPbbeoL4TYe0nFAZ/fZYaLZM17cvGXR5x/lH7CL8B/QwF+6avtyNrsrSartLLWet8C/Xw5dfFF31TrYeP0gHuih6hgG33gEh7uMl0XXbWiXwrBsL1V3oMb+XCIcOfXyPSqPrZ3/I5ggYb0P7UykNNqOtHcw6VD5I6P61p1kRLuq2yOqEgfL82hKG1cfU0jIc+zFlOTZRr7M4YnGxntJpV9Jf+LA3Ps+m5XpP5OBR28FmI9c5rn2+pHyBVl/u2cZVUykyoyJeZQPer9srOMJZvj/4knQZFOkYkl64xuTy4knZFvvhQD/sRHjIKJGOITmVvXZc7OhdE2zjdCKcEWgsMcZqEdOQXAS9Y4Llbg/FhY4OGFpiEKdaxDQkl0E3TDABmwpHu+xCby2xRqQ1JKdj40pjz2En1Jpg+kamofdlWkusEWkNyXnkJjoqdY8p3QRzRH1gic0kbpe7KwzZBHOiW7qREPhudLIJ5pDpOL8hkBuNXjPWr+tUE8wlM7DEQvDLuxhOn/Ffn4F5TzLBxmT6llgI9HGNkwk2HqOwM17aBJvxNuMm2Azx09JjjcB/l+BaeZnkxwIAAAAASUVORK5CYII=)
On integrating both sides, we get,
------(1)
Let ex = t
⇒ e2x = t2
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ exdx = dt
Substituting the value in equation (1), we get,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK0AAAAtCAMAAADrwRupAAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OgBmOjoAOjpmOmY6OmZmOmaQOma2OpCQOpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLyQkLbbkNv/tmYAtmY6tmZmtpA6ttv/tv//25A625Bm27Zm27aQ29uQ29u22/+22////7Zm/9uQ/9u2//+2///bEVyXVQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC20lEQVRYR+1Yi3LaMBC0aNKoz5DGgb5w0gbThwJtkbD+/896OpdYsjhbBjuobTSMxwMGrZbV7Z6S5HH8OwxIxmZ/0WpkZGj1grFLkj/FgVv1bB4LwfllAi9qIFo5igWtev6dRDph7BWgBfIZe0I+9pC056drYjrFPyb6xug2Gm51RqoA14G7LBq0ilNoi9R8Epdu6QoVI1pB1tMo0ZLFSWdnoIS4dJvTpVSMbpPN1HBvxP1lmJKrF4aTwKEzuo6CyY3GP7iBC3fDxIW715Oe0AYu+MDHRAe0RXpsj+qGlrSy/TlbgWOPzsFaqDRUpMbLyzBKo9VZXX1F2j9awd6s9Veze0M8sAtaxXtHq7hRor6eb9OQXr5gbGyKIZvdcXNnDZ0/xZyiM8ZO14qzqkb53A6AVlTzIbdQt9caC6XiL+eJa0fCCAK9X2cYS6yC6qOVrMOWDFKyXROructQX16IYCLMfrd1MQDavNwvMP6wYu+EXWhLQ98xzFqMfnB4P4vv2tzezxt0QxHto9WrKcf9j9ySaI3TS3sXDcCtB9pTgtHtVgRNaBOQQm7XLB+taGggg2Tq/2VW8kAl4KXSLcktsH7x1m6nBkDrL0kiAahATEOf4bJ5h83oNiERPOQudQ+CVmcj6OfQHco0BE3o+FfKZlBM2czYGNWtSDcF7ELbPVv9BGN1S7zLlV4ArJP3VHNKC0y2VdO81jqspq0xtri6TZa9HzCAcu7LF7keN4xvJrVjA/fPqEJK/4c3OhvftFGbOGjV+Tc0lGq4aKtiL+jDnaBK4j8EOcGktsbhtQ5NaKsjm2WTbPeEG/C1MkpYI4hbcRywUJBrWglBKwGs7F5KAshrecQzFgut71Klbsvgfwy0aIidlXA4S/v9Qs09IF+G1YT9Zjv0W6Le58SMdocTuyZVc4fhjmxaaS+u5k74hdz0CQyenXyg3GG4I5tWsGZv997vts/6+MR/yMBv7m9ShNb+ahoAAAAASUVORK5CYII=)
⇒ tan-1 y + tan-1 t = C
⇒ tan-1 y + tan-1 (ex) = C -------(2)
Now, y =1 at x = 0
Therefore, equation (2) becomes:
tan-1 1 + tan-1 1 = C
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Substituting
in (2), we get,
tan-1 y + tan-1 (ex) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAeCAMAAADJnMQBAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAABmADqQAGa2OgAAOjoAOmaQOma2ZgAAZjoAZjpmZrbbZrb/kDoAkDo6kGaQtmYAtmZmttv/tv//25A629uQ2////7Zm/9uQ/9vb//+2///b/nd0/AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAY0lEQVQoU5WOSRKAIAwExwVXghsoCv9/pwE8WVzo01QmqTSQwU1VhLAfG8G0cSeTvArtcwFuJFjRcyom/WKKL/8Ht6A08kv3JSNZMWCHIMu4WXslVzY18WuTnL+W0ylqnVV5AWUEBCjvGO+3AAAAAElFTkSuQmCC)
Question 15.Solve the differential equation ![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
------(1)
Let ![](data:image/png;base64,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)
Differentiating it w.r.t. y, we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
------(2)
From equation (1) and equation (2), we get,
![](data:image/png;base64,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)
⇒ dz = dy
On integrating both sides, we get,
z = y + C
![](data:image/png;base64,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)
Question 16.Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = –1, when x = 0. (Hint: put x – y = t)
Answer:It is given that (x – y)(dx +dy) = dx – dy
⇒ (x –y + 1)dy = (1- x + y)dx
![](data:image/png;base64,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)
----------------(1)
Let x – y = t
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, let us substitute the value of x-y and
in equation (1), we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
-------(2)
On integrating both side, we get,
t + log|t| = 2x + C
⇒ ( x – y ) + log |x – y| = 2x + C
⇒ log|x – y| = x + y + C --------(3)
Now, y = -1 at x = 0
Then, equation (3), we get,
log 1 = 0 -1 + C
⇒ C = 1
Substituting C = 1in equation (3), we get,
log|x – y| = x + y + 1
Therefore, a particular solution of the given differential equation is log|x – y| = x + y + 1.
Question 17.Solve the differential equation ![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p =
and Q =
)
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 18.Find a particular solution of the differential equation
(x ≠ 0), given that y = 0 when ![](data:image/png;base64,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)
Answer:It is given that ![](data:image/png;base64,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)
This is equation in the form of
(where, p = cotx and Q = 4xcosecx)
Now, I.F. = ![](data:image/png;base64,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)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
-----------(1)
Now, y = 0 at x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAeCAMAAADJnMQBAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6AGa2OgAAOgA6OmaQOma2OpDbZgAAZgA6ZjoAZrb/kDoAkLbbkNv/tmYAtmZmttv/25A627Zm29u22////7Zm/9uQ/9vb//+2///bvb40iwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAX0lEQVQoU5WPSRKAIAwEx100igqK4v//adgOVnGhT11JppIAGexceQj7sRFU42cy9i6u+1yAFQTT9mzFhF1McfIfOPnqwZXstELXMnbvLpkaY0n7MUYlMSyGXFaEx3N8HU8DSRzETU0AAAAASUVORK5CYII=)
Therefore, equation (1), we get,
0 = ![](data:image/png;base64,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)
⇒ C = ![](data:image/png;base64,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)
Now, substituting C =
in equation (1), we get,
ysinx = ![](data:image/png;base64,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)
Therefore, the required particular solution of the given differential equation is
ysinx = ![](data:image/png;base64,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)
Question 19.Find a particular solution of the differential equation
, given that y = 0 when x = 0.
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
On integrating both sides, we get,
----------------(1)
Let ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ eydt = -dt
Substituting value in equation (1), we get,
![](data:image/png;base64,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)
⇒ -log|t| = log|C(x+1)|
⇒ -log|2 – ey| = log|C(x + 1)|
![](data:image/png;base64,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)
------------------(2)
Now, at x = 0 and y = 0, equation (2) becomes,
![](data:image/png;base64,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)
⇒ C = 1
Now, substituting the value of C I equation (2), we get,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the required particular solution of the given differential equation is
![](data:image/png;base64,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)
Question 20.The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?
Answer:Let the population at any instant (t) be y.
Now it is given that the rate of increase of population is proportional to the number of inhabitants at any instant.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Now, integrating both sides, we get,
log y = kt + C ------(1)
According to given conditions,
In the year 1999, t = 0 and y = 20000
⇒ log20000 = C ----(2)
Also, in the year 2004, t = 5 and y = 25000
⇒ log 25000 = k.5 + C
⇒ log 25000 = 5k + log 20000
![](data:image/png;base64,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)
--------(3)
Also, in the year 2009, t = 10
Now, substituting the values of t, k and c in equation (1), we get
logy = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ y = 31250
Therefore, the population of the village in 2009 will be 31250.
Question 21.The general solution of the differential equation
is
A. xy = C
B. x = Cy2
C. y = Cx
D. y = Cx2
Answer:It is given that ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIgAAAAuCAMAAADEIaGTAAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOjqQOmY6OmaQOma2OpCQOpDbZgAAZjoAZjo6ZjpmZmYAZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm29uQ2////7Zm/9uQ/9u2//+2///b/mLDqAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACDklEQVRYR+1X0VbCMAxNESaiU8SBMnVKt/X/P9FkXcdkLaQOWh/IA+d02W0vSZb0AlztjxGQQrx4Qb0B3N2lJxHwBjCZlIlfRMAb8N+JqEyI6a5MxGQL5aMQc4xITo8KV7X0vDwAMwQqm+4AJPFIVqA2VCM5rlTmylHn5QKYTIqbT4Bihsc3jIgIkSseXPjOywUwiVC1qfUW6pSO1rVXJvOFgWMqWsM4NdZ6XQDmuYPXVDYDicHo7wuFOdS6q/Z6AHjcMDc5hqG/b718poQ5rPXyATweyOB+iadSZEyNrLd1SiurYR4bLxvA5IFVKprCLCZvUD1hsaoc11I/HNreywSweYDUaVDvYrL4TsQqw6ZSp8LOBBuP8fIAfB4gnVnw2GTsq5hw+nbjm8oWmwABUa9C3H4c+7uY8jtsqBc2asZVZukIX56zfixPGmXWm8ulLhEuwjkFw9qajrfwsQE4xOsJX6fNIOlG18EEO/ehuN9gTGoi+ndIsi2dQ4Kj1q6hwCFygYhY/rVODfaJuKmBtlgt0ytssUJB9z6b8Aj9+dIHY63V0A0NKhQIATp4iIK/nvE7AkbV4aWt04NxYmRU3V4PxuGx13ydHoxEpNN8Rg/G4kEio7nRGj0YjUin+Vo9GIuIUXU0Nhs9GMn6mq/Vg1GY9FQdDs8jsjwou6se7Ic7kB48neEwevA0j/O98QMz8S82hJGLTAAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Integrating both sides, we get,
log|x| - log|y| = log k
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ y = Cx where C = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAADlQTFRFAAAAAAAAADqQAGaQAGa2OgAAOmZmOpDbZjoAZma2tmYAtmY625A625Bm29v/2////7Zm/9uQ///bAcSIGwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAU0lEQVQYV42P2w6AMAhDmcMbMp38/8eOEt+Ixr40IaUciJLsYMzasoUT6beb1Aup4lpz3csEadfvvJaHQjKNoEfZ2n4GxxQ8fM+I+jfddzvgJN0b2cIB740OOFsAAAAASUVORK5CYII=)
Question 22.The general solution of a differential equation of the type
is
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:The integrating factor of the given differential equation
is ![](data:image/png;base64,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)
Thus, the general solution of the differential equation is given by,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 23.The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
A. x ey + x2 = C
B. x ey + y2 = C
C. y ex + x2 = C
D. y ey + x2 = C
Answer:It is given that exdy + (yex + 2x) dx = 0
![](data:image/png;base64,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)
![](data:image/png;base64,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)
This is equation in the form of
(where, p = 1 and Q = -2xe-x)
Now, I.F. = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIsAAAAZCAMAAAA2aS2FAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjqQOmaQOma2OpDbZgAAZgA6ZgBmZjoAZjqQZpC2ZpDbZrbbZrb/kDoAkGYAkLb/kNv/tmYAtmY6tpA6ttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///beDdkNAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABiklEQVRIS+2V23aCMBBFJ6FV0oultVVpoWqwpAj//30dCDcdwAT6yDy5XDl7Tk5mDQBz3U5AskV8+9TAicmAmp2s0vdJXiYDmmvKl+srS+ba5NQCWCpJl3BH/7Ly0gaEVsrrxplPvUgb4gXASkkySL3Ci2LLPeMbgPOaMXyjk+BBIkyGugRYK8sObUMVSjoBnHiQ+W6cfWIuahHLlclMVwBrJe1Qe8EM8HfyEAAUScs7TMmgKoC9knTo9RI9mrwQ5BcoHDdeTJXVuWRdTAeiXjGIPAbnAJIHqbeKz/4yht+vzL8/9OdCANbKqkMitvAj8gulb0ftBUeW4x9KsOc920TCzcKBhUwBtsqyA0CI+eOw5dE+6QGVQy8SolFdXKfYAeiMkApbSq1IvXzZoiG1U+Xa/TabjrJjB8BgzC9aV+dTr7ikc6w/aypfKuZFAabaWtl4IR+hQRaJWudyu+gbEaUelQk1HkCVim0hi8yWWafl8QCqjATjHyYrvi+88YDxygkPOUvnBP49gT8A7yuri+MpAAAAAABJRU5ErkJggg==)
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ yex = -x2 + C
⇒ yex + x2 = C
For each of the differential equations given below, indicate its order and degree (if defined).
Answer:
It is given that equation is
We can see that the highest order derivative present in the differential is .
Thus, its order is two. It is polynomial equation in . The highest power raised to
is 1.
Therefore, its degree is one.
Question 2.
For each of the differential equations given below, indicate its order and degree (if defined).
Answer:
It is given that equation is
We can see that the highest order derivative present in the differential is.
Thus, its order is one. It is polynomial equation in . The highest power raised to
is 3.
Therefore, its degree is three.
Question 3.
For each of the differential equations given below, indicate its order and degree (if defined).
Answer:
It is given that equation is
We can see that the highest order derivative present in the differential is .
Thus, its order is four. The given differential equation is not a polynomial equation.
Therefore, its degree is not defined.
Question 4.
For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
xy = a ex + b e–x + x2 :
Answer:
It is given that xy = a ex + b e–x + x2
Now, differentiating both sides w.r.t. x, we get,
Now, Again differentiating both sides w.r.t. x, we get,
Now, Substituting the values of ’ and
in the given differential equations, we get,
LHS =
= x(aex +be-x + 2) + 2(aex - be-x + 2) –x(aex +be-x + x2) + x2 – 2
= (axex +bxe-x + 2x) + 2(aex - be-x + 2) –x(aex +be-x + x2) + x2 – 2
= 2aex -2be-x +x2 + 6x -2
≠ 0
⇒ LHS ≠ RHS.
Therefore, the given function is not the solution of the corresponding differential equation.
Question 5.
For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
y = ex (a cos x + b sin x) :
Answer:
It is given that y = ex(acosx + bsinx) = aexcosx + bexsinx
Now, differentiating both sides w.r.t. x, we get,
Now, again differentiating both sides w.r.t. x, we get,
Now, Substituting the values of ’ and
in the given differential equations, we get,
LHS =
=2ex(bcosx – asinx) -2ex[(a + b)cosx + (b –a ) sinx] + 2ex(acosx + bsinx)
=ex[(2bcosx – 2asinx) - (2acosx + 2bcosx) - (2bsinx – 2asinx) + (2acosx + 2bsinx)]
= ex[(2b – 2a – 2b + 2a)cosx] + ex[(-2a – 2b + 2a + 2bsinx]
= 0 = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question 6.
For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
y = x sin 3x :
Answer:
It is given that y = xsin3x
Now, differentiating both sides w.r.t. x , we get,
Now, again differentiating both sides w.r.t. x, we get,
Now, substituting the value of in the LHS of the given differential equation, we get,
= (6.cos3x – 9xsin3x) + 9xsin3x – 6cos3x
= 0 = RHS
Therefore, the given function is the solution of the corresponding differential equation.
Question 7.
For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
x2 = 2y2 log y :
Answer:
It is given that x2 = 2y2 log y
Now, differentiating both sides w.r.t. x, we get,
2x = 2.
Now, substituting the value of in the LHS of the given differential equation, we get,
= xy –xy
= 0
Therefore, the given function is the solution of the corresponding differential equation.
Question 8.
Form the differential equation representing the family of curves given by (x – a)2 + 2y2 = a2, where a is an arbitrary constant.
Answer:
It is given that (x – a)2 + 2y2 = a2
⇒ x2 + a2 – 2ax + 2y2 = a2
⇒ 2y2 = 2ax – x2 ---------(1)
Now, differentiating both sides w.r.t. x, we get,
- ---------(2)
So, equation (1), we get,
2ax = 2y2 + x2
On substituting this value in equation (3), we get,
Therefore, the differential equation of the family of curves is given as .
Question 9.
Prove that x2 – y2 = c (x2 + y2)2 is the general solution of differential equation (x3–3xy2) dx = (y3–3x2y)dy, where c is a parameter.
Answer:
It is given that (x3–3xy2) dx = (y3–3x2y)dy
- --------(1)
Now, let us take y = vx
Now, substituting the values of y and in equation (1), we get,
On integrating both sides we get,
--------(2)
Now,
---------(3)
Let
⇒
⇒
⇒
Now,
Let v2 = p
Now, substituting the values of I1 and I2 in equation (3), we get,
Thus, equation (2), becomes,
⇒ (x2 – y2)2 = C’4(x2 + y2 )4
⇒ (x2 – y2) = C’2(x2 + y2 )
⇒ (x2 – y2) = C(x2 + y2 ), where C = C’2
Therefore, the result is proved.
Question 10.
Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.
Answer:
We know that the equation of a circle in the first quadrant with centre (a, a) and radius a which touches the coordinate axes is :
(x -a)2 + (y –a)2 = a2 -----------(1)
Now differentiating above equation w.r.t. x, we get,
2(x-a) + 2(y-a) = 0
⇒ (x – a) + (y – a)y’ = 0
⇒ x – a +yy’ – ay’ = 0
⇒ x + yy’ –a(1+y’) = 0
⇒ a =
Now, substituting the value of a in equation (1), we get,
⇒ (x - y)2.y’2 + (x – y)2 = (x + yy’)2
⇒ (x – y)2[1 + (y’)2] = (x + yy’)2
Therefore, the required differential equation of the family of circles is
(x – y)2[1 + (y’)2] = (x + yy’)2
Question 11.
Find the general solution of the differential equation
Answer:
It is given that
On integrating, we get,
⇒ sin-1x + sin-1y = C
Question 12.
Show that the general solution of the differential equation is given by (x + y + 1) = A (1 – x – y – 2xy), where A is parameter.
Answer:
It is given that
On integrating both sides, we get,
Let
Then,
Now, let A = , then, we have,
Question 13.
Find the equation of the curve passing through the point whose differential equation is sin x cos y dx + cos x sin y dy = 0.
Answer:
It is given that sin x cos y dx + cos x sin y dy = 0
⇒ tanxdx + tanydy = 0
So, on integrating both sides, we get,
log(secx) + log(secy) = logC
⇒ log(secx.secy) = log C
⇒ secx.secy = C
The curve passes through point
Thus, 1× = C
⇒ C =
On substituting C = in equation (1), we get,
secx.secy =
Therefore, the required equation of the curve is
Question 14.
Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.
Answer:
It is given that (1 + e2x) dy + (1 + y2) ex dx = 0
On integrating both sides, we get,
------(1)
Let ex = t
⇒ e2x = t2
⇒ exdx = dt
Substituting the value in equation (1), we get,
⇒ tan-1 y + tan-1 t = C
⇒ tan-1 y + tan-1 (ex) = C -------(2)
Now, y =1 at x = 0
Therefore, equation (2) becomes:
tan-1 1 + tan-1 1 = C
Substituting in (2), we get,
tan-1 y + tan-1 (ex) =
Question 15.
Solve the differential equation
Answer:
It is given that
------(1)
Let
Differentiating it w.r.t. y, we get,
------(2)
From equation (1) and equation (2), we get,
⇒ dz = dy
On integrating both sides, we get,
z = y + C
Question 16.
Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = –1, when x = 0. (Hint: put x – y = t)
Answer:
It is given that (x – y)(dx +dy) = dx – dy
⇒ (x –y + 1)dy = (1- x + y)dx
----------------(1)
Let x – y = t
Now, let us substitute the value of x-y and in equation (1), we get,
-------(2)
On integrating both side, we get,
t + log|t| = 2x + C
⇒ ( x – y ) + log |x – y| = 2x + C
⇒ log|x – y| = x + y + C --------(3)
Now, y = -1 at x = 0
Then, equation (3), we get,
log 1 = 0 -1 + C
⇒ C = 1
Substituting C = 1in equation (3), we get,
log|x – y| = x + y + 1
Therefore, a particular solution of the given differential equation is log|x – y| = x + y + 1.
Question 17.
Solve the differential equation
Answer:
It is given that
This is equation in the form of (where, p =
and Q =
)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
Question 18.
Find a particular solution of the differential equation (x ≠ 0), given that y = 0 when
Answer:
It is given that
This is equation in the form of (where, p = cotx and Q = 4xcosecx)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
-----------(1)
Now, y = 0 at x =
Therefore, equation (1), we get,
0 =
⇒ C =
Now, substituting C = in equation (1), we get,
ysinx =
Therefore, the required particular solution of the given differential equation is
ysinx =
Question 19.
Find a particular solution of the differential equation , given that y = 0 when x = 0.
Answer:
It is given that
On integrating both sides, we get,
----------------(1)
Let
⇒ eydt = -dt
Substituting value in equation (1), we get,
⇒ -log|t| = log|C(x+1)|
⇒ -log|2 – ey| = log|C(x + 1)|
------------------(2)
Now, at x = 0 and y = 0, equation (2) becomes,
⇒ C = 1
Now, substituting the value of C I equation (2), we get,
Therefore, the required particular solution of the given differential equation is
Question 20.
The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?
Answer:
Let the population at any instant (t) be y.
Now it is given that the rate of increase of population is proportional to the number of inhabitants at any instant.
Now, integrating both sides, we get,
log y = kt + C ------(1)
According to given conditions,
In the year 1999, t = 0 and y = 20000
⇒ log20000 = C ----(2)
Also, in the year 2004, t = 5 and y = 25000
⇒ log 25000 = k.5 + C
⇒ log 25000 = 5k + log 20000
--------(3)
Also, in the year 2009, t = 10
Now, substituting the values of t, k and c in equation (1), we get
logy =
⇒ y = 31250
Therefore, the population of the village in 2009 will be 31250.
Question 21.
The general solution of the differential equation is
A. xy = C
B. x = Cy2
C. y = Cx
D. y = Cx2
Answer:
It is given that
Integrating both sides, we get,
log|x| - log|y| = log k
⇒
⇒
⇒
⇒ y = Cx where C =
Question 22.
The general solution of a differential equation of the type is
A.
B.
C.
D.
Answer:
The integrating factor of the given differential equation
is
Thus, the general solution of the differential equation is given by,
Question 23.
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
A. x ey + x2 = C
B. x ey + y2 = C
C. y ex + x2 = C
D. y ey + x2 = C
Answer:
It is given that exdy + (yex + 2x) dx = 0
This is equation in the form of (where, p = 1 and Q = -2xe-x)
Now, I.F. =
Thus, the solution of the given differential equation is given by the relation:
y(I.F.) =
⇒ yex = -x2 + C
⇒ yex + x2 = C