$\frac{d y}{d x}+y=2 e^{2 x}$ then $y$ is

$(1+\mathrm{x}) \mathrm{e}^{-\mathrm{x}}+\frac{2}{3} \mathrm{e}^{2 \mathrm{x}}+\mathrm{c}$

$\mathrm{ce}^{-\mathrm{x}}+\frac{2}{3} \mathrm{e}^{2 \mathrm{x}}+\mathrm{c}$

$\mathrm{e}^{-\mathrm{x}}+\frac{2}{3} \mathrm{e}^{2 \mathrm{x}}+\mathrm{c}$

The solution $y^{n-1}\left(a y^{\prime}+y\right)=x$ is

$(n-1) y^{n-1}=C e^{-\frac{n x}{a}}+n x+a$

$n y^{n-1}=C e^{-\frac{n x}{a}}+n x$

$n y^{n-1}=C e^{-\frac{n x}{a}}+n x+a$

The solution of differential equation $\left( {1 + x} \right)y{\rm{ }}dx + \left( {1 - y} \right)x{\rm{ }}dy = 0$ is

${\log _e}\left( {\frac{x}{y}} \right) + x + y = c$

${\log _e}\left( {\frac{x}{y}} \right) - x + y = c$

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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Medium

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 421-450 of 452 questions

421 Medium

421 of 452

Application +4 | -1

\frac{d y}{d x}+y=2 e^{2 x}

then

y

View answer and explanation

Correct option: A

\operatorname{ce}^{-\mathrm{x}}+\frac{2}{3} \mathrm{e}^{2 \mathrm{x}}

Option A (

ce^{-\mathrm{x}}+\frac{{2}}{{3}} \mathrm{e}^{2 \mathrm{x}}

) is correct because it is the general solution to the given first-order linear differential equation.

422 Medium

422 of 452

Analyzing +4 | -1

2\left(x-\frac{y d y}{d x}\right)\left(x^2+y^2\right)=\left(x^2-y^2\right)\left(y-\frac{x d y}{d x}\right), y(1)=0

, then

View answer and explanation

Correct option: C

\ln \left(x^2-y^2\right)+\tan ^{-1} \frac{y}{x}=0

Option C,

\ln \left(x^2-y^2\right)+\tan ^{-1} \frac{y}{x}=0

, is correct because it is the solution to the given differential equation with the initial condition y(1)=0.

423 Medium

423 of 452

Application +4 | -1

One of the solution of differential equation

\frac{d y}{d x}=\frac{y}{y \sin y-x}

View answer and explanation

Correct option: A

y(x+\cos y)=2022+\sin y

Option A (

y(x+\cos y)=2022+\sin y

) is a solution because it satisfies the given differential equation after rearranging and integrating.

424 Medium

424 of 452

Application +4 | -1

The solution of the differential equation

x \frac{d y}{d x}=y(\log y-\log x+1)

View answer and explanation

Correct option: D

\log \left(\frac{y}{x}\right)= c x

Option D (

\log \left(\frac{y}{x}\right)= c x

) is the correct solution, found by recognizing the equation as homogeneous and using the appropriate substitution.

425 Medium

425 of 452

Application +4 | -1

The solution of the differential equation

\frac{d y}{d x}-e^{y \mid x}+e^{y-x}

is ......

View answer and explanation

Correct option: A

e^{-y}=e^{-x} - e^x+c

Option A (

e^{-y} = e^{-x} - e^x + c

) is correct because it is the solution obtained by separating variables and integrating the given differential equation.

426 Medium

426 of 452

Application +4 | -1

The solution of

y^{\prime}=(x+y)^2, y(0)=0

is given by

View answer and explanation

Correct option: A

\tan x=x+y

Option A (tan x = x+y) is correct because it is the solution obtained by substituting u = x+y and integrating the resulting separable equation with the given initial condition.

427 Medium

427 of 452

Application +4 | -1

Solution of the equation

x d y=\left(y+x \frac{f\left(\frac{y}{x}\right)}{f^{\prime}\left(\frac{y}{x}\right)}\right) d x

View answer and explanation

Correct option: C

f\left(\frac{y}{x}\right)=c x

Option C,

f\left(\frac{y}{x}\right)=c x

, is the correct solution obtained by solving the given homogeneous differential equation using substitution and integration.

428 Medium

428 of 452

Application +4 | -1

The solution of

\frac{d y}{d x}=\frac{y}{x}-\sin \left(\frac{y}{x}\right)

View answer and explanation

Correct option: C

y=2 x \tan ^{-1}\left(\frac{c}{x}\right)

Option C (

y=2 x \tan ^{-1}\left(\frac{c}{x}\right)

) is correct because it is the solution obtained by solving the homogeneous differential equation using the substitution

y=vx

429 Medium

429 of 452

Application +4 | -1

The solution of

y e^{-x / y} d x-\left(x e^{-x / y}+y^3\right) d y=0

View answer and explanation

Correct option: A

\frac{y^2}{2}+e^{-x / y}=k

Option A (

\frac{y^2}{2}+e^{-x / y}=k

) is the correct solution, found by rearranging the equation and integrating using appropriate substitutions.

430 Medium

430 of 452

Application +4 | -1

A curve passing through origin, all the normals to which pass through

(1,2)

View answer and explanation

Correct option: D

a circle with centre

(1,2)

Option D (a circle with centre (1,2)) is correct because the equation derived from the given conditions represents a circle centered at (1,2).

431 Medium

431 of 452

Application +4 | -1

If the solution curve of the differential equation

\frac{d y}{d x}=\frac{x+y-2}{x-y}

passes through the points

(2,1)

and

(k+1,2), k>0

, then

View answer and explanation

Correct option: A

2 \tan ^{-1}\left(\frac{1}{k}\right)=\log _e\left(k^2+1\right)

Option A (

2 \tan^{-1}(\frac{{1}}{{k}}) = \log_e(k^2+1)

) is correct because after solving the homogeneous differential equation and applying the given points, this relationship is obtained.

432 Medium

432 of 452

Application +4 | -1

The solution of the differential equation

\frac{d y}{d x}+\frac{x y}{1-x^2}=x \sqrt{y},(|x|<1)

\sqrt{y}=-\frac{f(x)}{3}+C\left(1-x^2\right)^{\frac{1}{4}}

, where

f\left(\frac{1}{2}\right)=\frac{3}{4}

and

C

is an arbitrary constant. Then, the value of

f\left(-\frac{1}{2}\right)

View answer and explanation

Correct option: B

\frac{3}{4}

Option B (

\frac{3}{4}

) is correct because it is the value of the function

f(x)

x=-\frac{1}{2}

, where

f(x)

is derived from solving the given differential equation.

433 Medium

433 of 452

Application +4 | -1

The solution of

y^{\prime} \frac{2 x y}{1-x^2}=1+x

\left.y\right|_{x=0}=1

is given by

View answer and explanation

Correct option: D

y=\frac{1}{1-x^2}\left(1+x+\frac{x^2}{2}-\frac{x^3}{3}-\frac{x^4}{4}\right)

The correct answer is D (

y=\frac{{1}}{{1-x^2}}\left(1+x+\frac{{x^2}}{{2}}-\frac{{x^3}}{{3}}-\frac{{x^4}}{{4}}\right)

) because it is the solution of the linear differential equation with the given initial condition.

434 Medium

434 of 452

Application +4 | -1

A differential equation representing the family of parabolas with axis parallel to

y

-axis and whose length of latusrectum is the distance from the point origin to the line

4 x-3 y+5=0

is given by

View answer and explanation

Correct option: B

\frac{\mathrm{d}^2 y}{\mathrm{dx}^2}=2

Option B,

\frac{\mathrm{d}^2 y}{\mathrm{dx}^2}=2

, is the correct differential equation representing the family of parabolas with the given conditions.

435 Medium

435 of 452

Application +4 | -1

The general solution of the differential equation

\sec ^2 x \tan y d x+\sec ^2 y \tan x d y=0

x, y \in\left(0, \frac{\pi}{4}\right)

View answer and explanation

Correct option: B

\cot x \cot y=c

Option C (

\cot x \cot y = c

) is incorrect. The correct answer is

\tan x \tan y = c

because it is the general solution obtained by separating variables and integrating the given differential equation.

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436 Medium

436 of 452

Application +4 | -1

The solution

y^{n-1}\left(a y^{\prime}+y\right)=x

View answer and explanation

Correct option: C

n y^n=C e^{-\frac{n x}{a}}+n x-a

Option C (

n y^n=C e^{-\frac{{n x}}{{a}}}+n x-a

) is correct because it is the solution obtained by making the substitution

u = y^n

and solving the resulting linear differential equation.

437 Medium

437 of 452

Application +4 | -1

Solution of the differential equation

y\left(a x y+e^x\right) d x-e^x d y=0

where

y(0)=1

View answer and explanation

Correct option: A

y\left(2-a x^2\right)=2 e^x

Option A (

y(2-ax^2)=2e^x

) is correct because it represents the solution to the given differential equation with the specified initial condition.

438 Medium

438 of 452

Application +4 | -1

The solution of

\frac{d y}{d x}+\frac{y}{3}=1

View answer and explanation

Correct option: B

y=3+c e^{-x / 3}

Option B (

y=3+c e^{-x / 3}

) is correct because it is the solution obtained by solving the given first-order linear differential equation using the integrating factor method.

439 Medium

439 of 452

Application +4 | -1

The solution of differential equation

\left( {1 + x} \right)y\mathrm{ }dx + \left( {1 - y} \right)x\mathrm{ }dy = 0

View answer and explanation

Correct option: A

{\log _e}\left( {xy} \right) + x - y = c

Option A (

{\log _e}\left( {xy} \right) + x - y = c

) is correct because it is the solution obtained by separating variables and integrating the given differential equation.

440 Medium

440 of 452

Application +4 | -1

A function

y = f\left( x \right)

has a second order derivative

f'' = 6\left( {x - 1} \right)

. If its graph passes through the point (2,1) and at point the tangent to the graph is

y = 3x - 5

then the function is

View answer and explanation

Correct option: B

{\left( {x - 1} \right)^3}

Option B (

{(x - 1)^3}

) is correct because it satisfies the given conditions for the second derivative, the point (2,1), and the tangent line

y = 3x - 5

441 Medium

441 of 452

Analyzing +4 | -1

The solution to the differential equation

\frac{{dy}}{{dx}} = \frac{{yf'\left( x \right) - {y^2}}}{{f\left( x \right)}}

where

f\left( x \right)

is a given function is

View answer and explanation

Correct option: A

f\left( x \right) = y\left( {x + c} \right)

Option A is correct. The solution to the differential equation is found by recognizing it as a Bernoulli equation and solving accordingly.

442 Medium

442 of 452

Application +4 | -1

The general solution of the differential equation

(x+y) d x+x d y=0

View answer and explanation

Correct option: C

x^2+2 x y=c

Option C (

x^2+2 x y=c

) is the correct general solution of the given differential equation, obtained by recognizing it as a homogeneous equation and applying the substitution

y=vx

443 Medium

443 of 452

Application +4 | -1

The solution of the differential equation

\frac{{dy}}{{dx}} - \frac{{\tan y}}{x} = \frac{{\tan y\sin y}}{{{x^2}}}

View answer and explanation

Correct option: A

\frac{x}{{\sin y}} + \log x = c

Option A (

\frac{x}{{\sin y}} + \log x = c

) is correct because it is the solution of the given differential equation after applying appropriate substitutions and integrating factors.

444 Medium

444 of 452

Application +4 | -1

y=0

when

x=1

, then a particular solution of the equation

\left(1+x^2\right) \frac{d y}{d x}+2 x y=\frac{1}{1+x^2}

View answer and explanation

Correct option: C

y\left(1+x^2\right)=\frac{\pi}{4}-\operatorname{Cot}^{-1} x

Option D (

y(1+x^2)=\operatorname{{Tan}}^{-1} x+\frac{{\pi}}{{4}}

) is incorrect. The correct answer is C (

y(1+x^2)=\frac{{\pi}}{{4}}-\operatorname{{Cot}}^{-1} x

). The error likely stems from incorrect handling of the initial condition or trigonometric identities.

445 Medium

445 of 452

Application +4 | -1

An integrating factor of the differential equation

\left(1-x^2\right) \frac{d y}{d x}-x y=1

, is

View answer and explanation

Correct option: C

\sqrt{\left(1-x^2\right)}

Option C (

\sqrt{\left(1-x^2\right)}

) is correct because it is the integrating factor obtained after rewriting the differential equation in standard form.

446 Medium

446 of 452

Application +4 | -1

The substitution

y=z^\alpha

transforms the differential equation

\left(x^2 y^2-1\right) d y+2 x y^3 d x=0

into a homogeneous differential equation for

View answer and explanation

Correct option: A

\alpha=-1

Option A is correct because the substitution transforms the given differential equation into a homogeneous equation when alpha is -1.

447 Medium

447 of 452

Application +4 | -1

The solution of the differential equation

y\left(\sin ^2 x\right) d y+(\sin x \cos x) y^2 d x=x d x

is (where

{C}

is the constant of integration)

View answer and explanation

Correct option: B

\sin ^2 x \cdot y^2=x^2+C

Option B (

\sin ^2 x \cdot y^2=x^2+C

) is correct because it is the solution obtained after rearranging and integrating the given differential equation.

448 Medium

448 of 452

Application +4 | -1

The tangent at point

(x, y)

on the curve

y=f(x)

makes an angle of measure

\tan ^{-1}(2 x+3 y)

with positive direction of

\mathrm{X}

-axis. If the curve is passing through the point

(1,2)

, then its equation is .......

View answer and explanation

Correct option: A

6 x+9 y+2=26 e^{3(x-1)}

Option B (

6x + 9y - 2 = 26e^{3(x-1)}

) is correct because it is the solution to the differential equation given the initial condition (1, 2).

449 Medium

449 of 452

Application +4 | -1

The differential equation whose solution is

y=c_1 \cos a x+c_2 \sin a x

is

(Where

c_1, c_2

are arbitrary constants)

View answer and explanation

Correct option: B

\frac{d^2 y}{d x^2}+a^2 y=0

Option B (

\frac{{d^2 y}}{{d x^2}}+a^2 y=0

) is correct because the second derivative of the given solution satisfies this differential equation.

450 Medium

450 of 452

Application +4 | -1

A curve is passing through the point

(0,1)

. If the slope of the curve at any point

(x, y)

is equal to

x+x y

, then the equation of the curve is

View answer and explanation

Correct option: B

y+1=2 e^{x^2 / 2}

Option B (

y+1=2 e^{x^2 / 2}

) is correct because it is the solution obtained after solving the given differential equation with the initial condition y(0)=1.

Revision notes and key takeaways

Review DIFFERENTIAL EQUATIONS in your textbook or class notes, then use this set to check your understanding.

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