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# NCERT Solutions for Class 12 Maths Chapter 9 - Exercise LIVE
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## NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations (Ex 9.6) Exercise 9.6

Free PDF download of NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.6 (Ex 9.6) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 12 Maths Chapter 9 Differential Equations Exercise 9.6 Questions with Solutions to help you to revise complete Syllabus and Score More marks. Register and get all exercise solutions in your emails.

NCERT solutions class 12 maths chapter nine exercise 9.6 differential equations facilitate children to study more about linear differential equations. There are  well-known forms of such equations, i.e., dy/dx + Py = Q and dx/dy + P1x = Q1. Given below are the steps to resolve such differential equations.

• Write the given differential equation in the form of dy/dx + Py = Q wherein P, Q are constants or functions of x only.

• Find the Integrating Factor (I.F), i.e., e ∫p dx

• Write the solution of the given differential equation as y (I.F) = ∫[Q (I.F.)] dx + C.

NCERT solutions class 12 maths chapter nine exercise 9.6 has a pool of nineteen problems requiring students to discover the general and the particular solution of a linear differential equation by using the above method.

The factor to remember while solving the NCERT solutions class 12 maths chapter nine exercise 9.6 differential equations is figuring out the right integrating factor. This is a new idea and students want to recognize a way to successfully use it in the problems.

Last updated date: 20th Sep 2023
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## Access NCERT Solutions for Class 12 Chapter 9 – Differential Equations

Exercise 9.6

1. Solve the differential equation $\dfrac{{dy}}{{dx}} + 2y = \sin x$.

Ans: The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = 2$ and $Q = \sin x$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$

$I.F. = {e^{\int {2dx} }}$

$I.F. = {e^{2x}}$

Therefore, solution of the given differential equation is given by the relation,

$y\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dx} + C$

$y{e^{2x}} = \int {\sin x.{e^{2x}}dx} + C.......\left( 1 \right)$

Let $I = \int {\sin x.{e^{2x}}dx}$

Using integration by parts,

$I = \sin x\int {{e^{2x}}dx} - \int {\left( {\dfrac{{d\left( {\sin x} \right)}}{{dx}}.\int {{e^{2x}}dx} } \right)} dx$

$I = \dfrac{{{e^{2x}}\sin x}}{2} - \int {\cos x.\dfrac{{{e^{2x}}}}{2}dx}$

Again using integration by parts,

$I = \dfrac{{{e^{2x}}\sin x}}{2} - \cos x\int {\dfrac{{{e^{2x}}}}{2}dx} + \int {\left( {\dfrac{{d\left( {\cos x} \right)}}{{dx}}.\int {\dfrac{{{e^{2x}}}}{2}dx} } \right)} dx$

$I = \dfrac{{{e^{2x}}\sin x}}{2} - \dfrac{{{e^{2x}}\cos x}}{4} + \int {\left( { - \sin x} \right).\dfrac{{{e^{2x}}}}{4}dx}$

$I = \dfrac{{{e^{2x}}\sin x}}{2} - \dfrac{{{e^{2x}}\cos x}}{4} - \dfrac{1}{4}\int {\sin x.{e^{2x}}dx}$

As $I = \int {\sin x.{e^{2x}}dx}$,

Therefore,

$I = \dfrac{{{e^{2x}}\sin x}}{2} - \dfrac{{{e^{2x}}\cos x}}{4} - \dfrac{1}{4}I$

$I + \dfrac{1}{4}I = \dfrac{{{e^{2x}}\sin x}}{2} - \dfrac{{{e^{2x}}\cos x}}{4}$

$\dfrac{5}{4}I = \dfrac{{{e^{2x}}\sin x}}{2} - \dfrac{{{e^{2x}}\cos x}}{4}$

$I = \dfrac{{{e^{2x}}}}{5}\left( {2\sin x - \cos x} \right)$

Substituting this value of I in equation 1.

$y{e^{2x}} = \dfrac{{{e^{2x}}}}{5}\left( {2\sin x - \cos x} \right) + C$

$y = \dfrac{1}{5}\left( {2\sin x - \cos x} \right) + C{e^{ - 2x}}$

This required differential equation .

2. Solve the differential equation $\dfrac{{dy}}{{dx}} + 3y = {e^{ - 2x}}$.

Ans: The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = 3$ and $Q = {e^{ - 2x}}$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$

$I.F. = {e^{\int {3dx} }}$

$I.F. = {e^{3x}}$

Therefore, solution of the given differential equation is given by the relation,

$y\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dx} + C$

$y{e^{3x}} = \int {\left( {{e^{3x}} \times {e^{ - 2x}}} \right)dx} + C$

$y{e^{3x}} = \int {{e^x}dx} + C$

$y{e^{3x}} = {e^x} + C$

$y = {e^{ - 2x}} + C{e^x}$

This required differential equation.

3. Solve the differential equation $\dfrac{{dy}}{{dx}} + \dfrac{y}{x} = {x^2}$.

Ans: The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = \dfrac{1}{x}$ and $Q = {x^2}$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$

$I.F. = {e^{\int {\dfrac{1}{x}dx} }}$

$I.F. = {e^{\log x}}$

$I.F. = x$

Therefore, solution of the given differential equation is given by the relation,

$y\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dx} + C$

$yx = \int {\left( {{x^2}.x} \right)dx} + C$

$yx = \int {\left( {{x^3}} \right)dx} + C$

$yx = \dfrac{{{x^4}}}{4} + C$

This required differential equation.

4. Solve the differential equation $\dfrac{{dy}}{{dx}} + \left( {\sec x} \right)y = \tan x{\text{ }}\left( {0 \leqslant x \leqslant \dfrac{\pi }{2}} \right)$

Ans: The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = \sec x$ and $Q = \tan x$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$

$I.F. = {e^{\int {\sec xdx} }}$

$I.F. = {e^{\log \left( {\sec x + \tan x} \right)}}$

$I.F. = \sec x + \tan x$

Therefore, solution of the given differential equation is given by the relation,

$y\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dx} + C$

$y\left( {\sec x + \tan x} \right) = \int {\tan x\left( {\sec x + \tan x} \right)dx} + C$

$y\left( {\sec x + \tan x} \right) = \int {\sec x\tan xdx} + \int {{{\tan }^2}xdx} + C$

$y\left( {\sec x + \tan x} \right) = \sec x + \int {\left( {{{\sec }^2}x - 1} \right)dx} + C$

$y\left( {\sec x + \tan x} \right) = \sec x + \tan x - x + C$

This required differential equation.

5. Solve the differential equation ${\cos ^2}x\dfrac{{dy}}{{dx}} + y = \tan x{\text{ }}\left( {0 \leqslant x\dfrac{\pi }{2}} \right)$

Ans: On rearranging the given equation,

$\dfrac{{dy}}{{dx}} + {\sec ^2}x.y = {\sec ^2}x\tan x$

The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = {\sec ^2}x$ and $Q = {\sec ^2}x\tan x$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$

$I.F. = {e^{\int {{{\sec }^2}xdx} }}$

$I.F. = {e^{\tan x}}$

Therefore, solution of the given differential equation is given by the relation,

$y\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dx} + C$

$y{e^{\tan x}} = \int {\tan x{{\sec }^2}x{e^{\tan x}}dx + C}$

Put $\tan x = t$

Differentiating w.r.t. $t$.

${\sec ^2}xdx = dt$

Therefore, above equation become,

$y{e^{\tan x}} = \int {t{e^t}dt + C}$

Using integration by parts,

$y{e^{\tan x}} = t{e^t} - \int {{e^t}dt + C}$

$y{e^{\tan x}} = t{e^t} - {e^t} + C$

Substituting the value of $t$ .

$y{e^{\tan x}} = \tan x{e^{\tan x}} - {e^{\tan x}} + C$

$y = \tan x - 1 + C{e^{ - \tan x}}$

This is required differential equation.

6. Solve the differential equation $x\dfrac{{dy}}{{dx}} + 2y = {x^2}\log x$

Ans: On rearranging given equation can be written as,

$\dfrac{{dy}}{{dx}} + \dfrac{2}{x}y = x\log x$

The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = \dfrac{2}{x}$ and $Q = x\log x$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$

$I.F. = {e^{\int {\dfrac{2}{x}dx} }}$

$I.F. = {e^{\log {x^2}}}$

$I.F. = {x^2}$

Therefore, solution of the given differential equation is given by the relation,

$y\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dx} + C$

$y{x^2} = \int {x\log x.{x^2}dx} + C$

$y{x^2} = \int {{x^3}\log xdx} + C$

Using integration by parts,

$y{x^2} = \log x\int {{x^3}dx} - \int {\left( {\dfrac{d}{{dx}}\left( {\log x} \right)\int {{x^3}dx} } \right)dx} + C$

$y{x^2} = \dfrac{{{x^4}\log x}}{4} - \int {\left( {\dfrac{1}{x}.\dfrac{{{x^4}}}{4}} \right)dx + C}$

$y{x^2} = \dfrac{{{x^4}\log x}}{4} - \dfrac{1}{4}.\dfrac{{{x^4}}}{4} + C$

$y{x^2} = \dfrac{{{x^4}\log x}}{4} - \dfrac{{{x^4}}}{{16}} + C$

$y = \dfrac{{{x^2}\log x}}{4} - \dfrac{{{x^2}}}{{16}} + C{x^{ - 2}}$

$y = \dfrac{{{x^2}}}{{16}}\left( {4\log x - 1} \right) + C{x^{ - 2}}$

This is required differential equation.

7. Solve the differential equation $x\log x\dfrac{{dy}}{{dx}} + y = \dfrac{2}{x}\log x$

Ans: On rearranging the given equation can be written as,

$\dfrac{{dy}}{{dx}} + \dfrac{y}{{\log x}} = \dfrac{2}{{{x^2}}}$

The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = \dfrac{1}{{x\log x}}$ and $Q = \dfrac{2}{{{x^2}}}$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$



$I.F. = {e^{\log \left( {\log x} \right)}}$

$I.F. = \log x$

Therefore, solution of the given differential equation is given by the relation,

$y\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dx} + C$

$y\log x = \int {\left( {\dfrac{2}{{{x^2}}}\log x} \right)dx} + C$

Using integration by parts,

$y\log x = 2\left[ {\log x\int {\dfrac{1}{{{x^2}}}dx} - \int {\left( {\dfrac{d}{{dx}}\left( {\log x} \right)\int {\dfrac{1}{{{x^2}}}dx} } \right)dx} } \right] + C$

$y\log x = 2\left[ {\log x\left( { - \dfrac{1}{x}} \right) - \int {\left( {\dfrac{1}{x}\left( { - \dfrac{1}{x}} \right)} \right)dx} } \right] + C$

$y\log x = 2\left[ {\log x\left( { - \dfrac{1}{x}} \right) + \int {\dfrac{1}{{{x^2}}}dx} } \right] + C$

$y\log x = 2\left[ {\log x\left( { - \dfrac{1}{x}} \right) - \dfrac{1}{x}} \right] + C$

$y\log x = - \dfrac{2}{x}\left( {1 + \log x} \right) + C$

This is required differential equation.

8. Solve the differential equation $\left( {1 + {x^2}} \right)dy + 2xydx = \cot xdx$

Ans: On rearranging the given equation can be written as,

$\dfrac{{dy}}{{dx}} + \dfrac{{2xy}}{{1 + {x^2}}} = \dfrac{{\cot x}}{{1 + {x^2}}}$

The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = \dfrac{{2x}}{{1 + {x^2}}}$ and $Q = \dfrac{{\cot x}}{{1 + {x^2}}}$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$

$I.F. = {e^{\int {\dfrac{{2x}}{{1 + {x^2}}}dx} }}$

$I.F. = {e^{\log \left( {1 + {x^2}} \right)}}$

$I.F. = 1 + {x^2}$

Therefore, solution of the given differential equation is given by the relation,

$y\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dx} + C$

$y\left( {1 + {x^2}} \right) = \int {\dfrac{{\cot x}}{{1 + {x^2}}} \times \left( {1 + {x^2}} \right)dx} + C$

$y\left( {1 + {x^2}} \right) = \int {\cot xdx} + C$

$y\left( {1 + {x^2}} \right) = \log \left( {\sin x} \right) + C$

This is required differential equation.

9. Solve the differential equation $x\dfrac{{dy}}{{dx}} + y - x + xy\cot x = 0$

Ans: On rearranging given equation can be written as,

$\dfrac{{dy}}{{dx}} + \left( {\dfrac{1}{x} + \cot x} \right)y = 1$

The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = \dfrac{1}{x} + \cot x$ and $Q = 1$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$

$I.F. = {e^{\int {\left( {\dfrac{1}{x} + \cot x} \right)dx} }}$

$I.F. = {e^{\log x + \log \sin x}}$

$I.F. = x\sin x$

Therefore, solution of the given differential equation is given by the relation,

$y\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dx} + C$

$yx\sin x = \int {\left( {1.x\sin x} \right)dx} + C$

$yx\sin x = x\int {\sin xdx - \int {\left( {\dfrac{d}{{dx}}\left( x \right)\int {\sin xdx} } \right)dx} } + C$

$yx\sin x = x\left( { - \cos x} \right) - \int {\left( { - \cos x} \right)dx} + C$

$yx\sin x = - x\cos x + \sin x + C$

$y = \dfrac{{ - x\cos x}}{{x\sin x}} + \dfrac{{\sin x}}{{x\sin x}} + \dfrac{C}{{x\sin x}}$

$y = - \cot x + \dfrac{1}{x} + \dfrac{C}{{x\sin x}}$

This is required differential equation.

10. Solve the differential equation $\left( {x + y} \right)\dfrac{{dy}}{{dx}} = 1$

Ans: On rearranging the given equation can be written as,

$\dfrac{{dx}}{{dy}} - x = y$

The given equation is in the form of $\dfrac{{dx}}{{dy}} + Px = Q$

Where, $P = - 1$ and $Q = y$

Calculating integration factor,

$I.F. = {e^{\int { - 1dy} }}$

$I.F. = {e^{ - y}}$

Therefore, solution of the given differential equation is given by the relation,

$x\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dy} + C$

$x{e^{ - y}} = \int {\left( {y{e^{ - y}}} \right)dy} + C$

$x{e^{ - y}} = y\int {{e^{ - y}}dy} - \int {\left( {\dfrac{d}{{dy}}\left( y \right)\int {{e^{ - y}}dy} } \right)dy} + C$

$x{e^{ - y}} = - y{e^{ - y}} - \int {\left( { - {e^{ - y}}} \right)dy} + C$

$x{e^{ - y}} = - y{e^{ - y}} - {e^{ - y}} + C$

$x = - y - 1 + C{e^y}$

$x + y + 1 = C{e^y}$

This is required differential equation.

11. Solve the differential equation $ydx + \left( {x - {y^2}} \right)dy = 0$

Ans: On rearranging the given equation can be written as,

$\dfrac{{dx}}{{dy}} + \dfrac{x}{y} = y$

The given equation is in the form of $\dfrac{{dx}}{{dy}} + Px = Q$

Where, $P = \dfrac{1}{y}$ and $Q = y$

Calculating integration factor,

$I.F. = {e^{\int {Pdy} }}$

$I.F. = {e^{\int {\dfrac{1}{y}dy} }}$

$I.F. = {e^{\log y}}$

$I.F. = y$

Therefore, solution of the given differential equation is given by the relation,

$x\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dy} + C$

$xy = \int {\left( {y.y} \right)dy} + C$

$xy = \int {{y^2}dy} + C$

$xy = \dfrac{{{y^3}}}{3} + C$

This is required differential equation.

12. Solve the differential equation $\left( {x + 3{y^2}} \right)\dfrac{{dy}}{{dx}} = y{\text{ }}\left( {y > 0} \right)$

Ans: On rearranging the given equation can be written as,

$\dfrac{{dx}}{{dy}} - \dfrac{x}{y} = 3y$

The given equation is in the form of $\dfrac{{dx}}{{dy}} + Px = Q$

Where, $P = - \dfrac{1}{y}$ and $Q = 3y$

Calculating integration factor,

$I.F. = {e^{\int {Pdy} }}$

$I.F. = {e^{\int {\left( { - \dfrac{1}{y}} \right)dy} }}$

$I.F. = {e^{\log \left( {\dfrac{1}{y}} \right)}}$

$I.F. = \dfrac{1}{y}$

Therefore, solution of the given differential equation is given by the relation,

$x\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dy} + C$

$x\left( {\dfrac{1}{y}} \right) = \int {\left( {3y \times \dfrac{1}{y}} \right)dy} + C$

$\dfrac{x}{y} = \int {3dy} + C$

$\dfrac{x}{y} = 3y + C$

$x = 3{y^2} + Cy$

This is required differential equation.

13. Solve the differential equation $\dfrac{{dy}}{{dx}} + 2y\tan x = \sin x;{\text{ }}y = 0{\text{ when }}x = \dfrac{\pi }{3}$

Ans: The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = 2\tan x$ and $Q = \sin x$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$

$I.F. = {e^{\int {2\tan xdx} }}$

$I.F. = {e^{\log \left( {{{\sec }^2}x} \right)}}$

$I.F. = {\sec ^2}x$

Therefore, solution of the given differential equation is given by the relation,

$y\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dx} + C$

$y{\sec ^2}x = \int {\left( {\sin x.{{\sec }^2}x} \right)dx} + C$

$y{\sec ^2}x = \int {\left( {\sec x\tan x} \right)dx} + C$

$y{\sec ^2}x = \sec x + C......\left( 1 \right)$

Now, $y = 0{\text{ when }}x = \dfrac{\pi }{3}$

$0 \times {\sec ^2}\left( {\dfrac{\pi }{3}} \right) = \sec \left( {\dfrac{\pi }{3}} \right) + C$

$C = - 2$

Substituting the value of $C = - 2$in equation $\left( 1 \right)$.

$y{\sec ^2}x = \sec x - 2$

$y = \cos x - 2{\sec ^2}x$

This is required differential equation.

14. Solve the differential equation $\left( {1 + {x^2}} \right)\dfrac{{dy}}{{dx}} + 2xy = \dfrac{1}{{1 + {x^2}}}{\text{ }}y = 0{\text{ when }}x = 1$

Ans: On rearranging the given equation can be written as,

$\dfrac{{dy}}{{dx}} + \dfrac{{2xy}}{{1 + {x^2}}} = \dfrac{1}{{{{\left( {1 + {x^2}} \right)}^2}}}$

The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = \dfrac{{2x}}{{1 + {x^2}}}$ and $Q = \dfrac{1}{{{{\left( {1 + {x^2}} \right)}^2}}}$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$

$I.F. = {e^{\int {\dfrac{{2x}}{{1 + {x^2}}}dx} }}$

$I.F. = {e^{\log \left( {1 + {x^2}} \right)}}$

$I.F. = 1 + {x^2}$

Therefore, solution of the given differential equation is given by the relation,

$y\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dx} + C$

$y\left( {1 + {x^2}} \right) = \int {\left( {\dfrac{1}{{{{\left( {1 + {x^2}} \right)}^2}}}.\left( {1 + {x^2}} \right)} \right)dx} + C$

$y\left( {1 + {x^2}} \right) = \int {\dfrac{1}{{1 + {x^2}}}dx} + C$

$y\left( {1 + {x^2}} \right) = {\tan ^{ - 1}}x + C......\left( 1 \right)$

Now, $y = 0{\text{ when }}x = 1$

$0 = {\tan ^{ - 1}}1 + C$

$C = - \dfrac{\pi }{4}$

Substituting the value of $C = - \dfrac{\pi }{4}$ in equation$\left( 1 \right)$,

$y\left( {1 + {x^2}} \right) = {\tan ^{ - 1}}x - \dfrac{\pi }{4}$

This is required differential equation.

15. Solve the differential equation $\dfrac{{dy}}{{dx}} - 3y\cot x = \sin 2x{\text{ }}y = 2{\text{ when }}x = \dfrac{\pi }{2}$

Ans: The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = - 3\cot x$ and $Q = \sin 2x$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$

$I.F. = {e^{\int { - 3\cot xdx} }}$

$I.F. = {e^{ - 3\log \left( {\sin x} \right)}}$

$I.F. = \dfrac{1}{{{{\sin }^3}x}}$

Therefore, solution of the given differential equation is given by the relation,

$y\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dx} + C$

$y.\dfrac{1}{{{{\sin }^3}x}} = \int {\left( {\sin 2x.\dfrac{1}{{{{\sin }^3}x}}} \right)dx} + C$

$y.\cos e{c^3}x = 2\int {\left( {\cot x\cos ecx} \right)dx} + C$

$y.\cos e{c^3}x = - 2\cos ecx + C$

$y = - \dfrac{2}{{\cos e{c^2}x}} + \dfrac{C}{{\cos e{c^3}x}}$

$y = - 2{\sin ^2}x + C{\sin ^3}x......\left( 1 \right)$

Now, $y = 2{\text{ when }}x = \dfrac{\pi }{2}$

$2 = - 2{\sin ^2}\left( {\dfrac{\pi }{2}} \right) + C{\sin ^3}\left( {\dfrac{\pi }{2}} \right)$

$2 = - 2 + C$

$C = 4$

Substituting the value of $C = 4$in equation 1,

$y = - 2{\sin ^2}x + 4{\sin ^3}x$

$y = 4{\sin ^3}x - 2{\sin ^2}x$

This is required differential equation.

16. Find the equation of a curve passing through the origin given that the slope of thetangent to the curve at any point $(x,y)$ is equal to the sum of the coordinates ofthe point.

Ans: Let $f\left( {x,y} \right)$ be the curve passing through the origin.

At point $(x,y)$, the slope of curve will be $\dfrac{{dy}}{{dx}}$ .

According to the question,

$\dfrac{{dy}}{{dx}} - y = x$

The given equation is in the form of $\dfrac{{dx}}{{dy}} + Px = Q$

Where, $P = - 1$ and $Q = x$

Calculating integration factor,

$I.F. = {e^{\int { - 1dx} }}$

$I.F. = {e^{ - x}}$

Therefore, solution of the given differential equation is given by the relation,

$x\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dy} + C$

$y{e^{ - x}} = \int {\left( {x{e^{ - x}}} \right)dx} + C$

$y{e^{ - x}} = x\int {{e^{ - x}}dy} - \int {\left( {\dfrac{d}{{dx}}\left( x \right)\int {{e^{ - x}}dx} } \right)dx} + C$

$y{e^{ - x}} = - x{e^{ - x}} - \int {\left( { - {e^{ - x}}} \right)dx} + C$

$y{e^{ - x}} = - x{e^{ - x}} - {e^{ - x}} + C$

$y = - x - 1 + C{e^x}$

$x + y + 1 = C{e^x}......\left( 1 \right)$

As, equation is passing through the origin.

Therefore,

$0 + 0 + 1 = C{e^0}$

$1 = C$

Substituting the value of $1 = C$ in equation 1.

$x + y + 1 = {e^x}$

This is required differential equation of curve passing through origin.

17. Find the equation of a curve passing through the point $\left( {0,2} \right)$ given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope ofthe tangent to the curve at that point by 5.

Ans: Let $f\left( {x,y} \right)$ be the curve passing through the origin.

At point $(x,y)$, the slope of curve will be $\dfrac{{dy}}{{dx}}$ .

According to the question,

$\dfrac{{dy}}{{dx}} + 5 = x + y$

$\dfrac{{dy}}{{dx}} - y = x - 5$

The given equation is in the form of $\dfrac{{dx}}{{dy}} + Px = Q$

Where, $P = - 1$ and $Q = x - 5$

Calculating integration factor,

$I.F. = {e^{\int { - 1dx} }}$

$I.F. = {e^{ - x}}$

Therefore, solution of the given differential equation is given by the relation,

$x\left( {I.F.} \right) = \int {\left( {Q \times I.F.} \right)dy} + C$

$y{e^{ - x}} = \int {\left( {x - 5} \right){e^{ - x}}dx} + C$

$y{e^{ - x}} = \left( {x - 5} \right)\int {{e^{ - x}}dx} - \int {\left( {\dfrac{d}{{dx}}\left( {x - 5} \right)\int {{e^{ - x}}dx} } \right)} dx + C$

$y{e^{ - x}} = \left( {5 - x} \right){e^{ - x}} + \int {{e^{ - x}}dx} + C$

$y{e^{ - x}} = \left( {5 - x} \right){e^{ - x}} - {e^{ - x}} + C$

$y{e^{ - x}} = \left( {4 - x} \right){e^{ - x}} + C$

$y = 4 - x + C{e^x}$

$y + x - 4 = C{e^x}......\left( 1 \right)$

As equation is passing through $\left( {0,2} \right)$ .

Therefore,

$0 + 2 - 4 = C{e^0}$

$- 2 = C$

Substituting the value of $- 2 = C$ in equation 1.

$y + x - 4 = - 2{e^x}$

This the required equation of curve passing through $\left( {0,2} \right)$ .

18. The Integrating Factor of the differential equation $x\dfrac{{dy}}{{dx}} - y = 2{x^2}$ is

$\left( A \right){e^{ - x}}$

$\left( B \right){e^{ - y}}$

$\left( C \right)\dfrac{1}{x}$

$\left( D \right)x$

Ans: On rearranging the given equation can be written as,

$\dfrac{{dy}}{{dx}} - \dfrac{y}{x} = 2x$

The given equation is in the form of $\dfrac{{dy}}{{dx}} + Py = Q$

Where, $P = - \dfrac{1}{x}$ and $Q = 2x$

Calculating integration factor,

$I.F. = {e^{\int {Pdx} }}$

$I.F. = {e^{\int { - \dfrac{1}{x}dx} }}$

$I.F. = {e^{ - \log x}}$

$I.F. = {e^{\log \dfrac{1}{x}}}$

$I.F. = \dfrac{1}{x}$

Therefore, the correct option is $\left( C \right)$ .

18. The Integrating Factor of the differential equation $\left( {1 - {y^2}} \right)\dfrac{{dx}}{{dy}} + yx = ay\left( { - 1 < y < 1} \right)$ is

$\left( A \right)\dfrac{1}{{{y^2} - 1}}$

$\left( B \right)\dfrac{1}{{\sqrt {{y^2} - 1} }}$

$\left( C \right)\dfrac{1}{{1 - {y^2}}}$

$\left( D \right)\dfrac{1}{{\sqrt {1 - {y^2}} }}$

Ans: On rearranging the given equation can be written as,

$\dfrac{{dx}}{{dy}} + \dfrac{{xy}}{{1 - {y^2}}} = \dfrac{{ay}}{{1 - {y^2}}}$

The given equation is in the form of $\dfrac{{dx}}{{dy}} + Px = Q$

Where, $P = \dfrac{y}{{1 - {y^2}}}$ and $Q = \dfrac{{ay}}{{1 - {y^2}}}$

Calculating integration factor,

$I.F. = {e^{\int {Pdy} }}$

$I.F. = {e^{\int {\dfrac{y}{{1 - {y^2}}}dy} }}$

$I.F. = {e^{\log \left( {\dfrac{1}{{\sqrt {1 - {y^2}} }}} \right)}}$

$I.F. = \dfrac{1}{{\sqrt {1 - {y^2}} }}$

Therefore, the correct option is $\left( D \right)$

## NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations Exercise 9.6

Opting for the NCERT solutions for Ex 9.6 Class 12 Maths is considered as the best option for the CBSE students when it comes to exam preparation. This chapter consists of many exercises. Out of which we have provided the Exercise 9.6 Class 12 Maths NCERT solutions on this page in PDF format. You can download this solution as per your convenience or you can study it directly from our website/ app online.

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