Differential equation of $y=\sec \left(\tan ^{-1} x\right)$ is

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

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

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

If integrating factor of $x\left( {1 - {x^2}} \right)dy + \left( {2{x^2}y - y - a{x^3}} \right)dx = 0$ is ${e^{\smallint Pdx}}$, then $P$ is equal to

$\frac{{2{x^2} - a{x^3}}}{{x\left( {1 - {x^2}} \right)}}$

$\frac{{2{x^2} - 1}}{{a{x^3}}}$

If integrating factor of $x\left( {1 - {x^2}} \right)dy + \left( {2{x^2}y - y - a{x^3}} \right)dx = 0{\rm{\,\, is \,\,}}{e^{\int P dx}}$ then $P$ is equal to

$\frac{2 x^2-a x^3}{x\left(1-x^2\right)}$

The solution of $(y+x+5) \mathrm{d} y=(y-x+$ 1) $\mathrm{d} x$ is

$\log \left((y+3)^2+(x+2)^2\right)+\tan ^{-1} \frac{y+3}{x+2}=C$

$\log \left((y+3)^2+(x-2)^2\right)+\tan ^{-1} \frac{y-3}{x-2}=C$

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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Mixed difficulty

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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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361 Easy

361 of 975

Application +4 | -1

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

, where

y(0)=0

, then

y

is expressed explicitly as

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it's the explicit solution to the given differential equation with the initial condition y(0)=0.

362 Easy

362 of 975

Application +4 | -1

c

is an arbitrary constant, then the general solution of the differential equation

y\mathrm{ }dx - x\mathrm{ }dy = xy\mathrm{ }dx

is given by

View answer and explanation

Correct option: D

y{e^x} = cx

Option B (

y\cos x = c + \frac{1}{2}\sin 2x

) is incorrect. Option D (

y\cos x = c + \frac{1}{2}\cos 2x

) is correct because it is the solution obtained after finding the integrating factor and integrating the differential equation.

363 Easy

363 of 975

Application +4 | -1

The solution of

\left(1+y+x^2 y\right) \mathrm{d} x+\left(x+x^3\right) \mathrm{d} y=0

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the solution obtained after solving the given differential equation.

364 Easy

364 of 975

Application +4 | -1

General solution of differential equation

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

View answer and explanation

Correct option: C

\sqrt{x^2+y^2}+\frac{y}{x}=c

Option C (

\sqrt{{x^2+y^2}}+\frac{{y}}{{x}}=c

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

365 Easy

365 of 975

Application +4 | -1

The integrating factor of

x\frac{{dy}}{{dx}} + \left( {1 + x} \right)y = x

View answer and explanation

Correct option: D

x{e^x}

Option D (

x{e^x}

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

366 Medium

366 of 975

Application +4 | -1

The solution of

2\left( {y + 3} \right) - xy\frac{{dy}}{{dx}} = 0

with

y = - 2

, where

x = 1

, is

View answer and explanation

Correct option: D

{x^2}{\left( {y + 3} \right)^3} = {e^{y + 2}}

The correct answer is D (

{x^2}{\left( {y + 3} \right)^3} = {e^{y + 2}}

) because it is the solution to the given differential equation

2\left( {y + 3} \right) - xy\frac{{dy}}{{dx}} = 0

with the initial condition

y = -2

when

x = 1

367 Easy

367 of 975

Application +4 | -1

Differential equation of

y=\sec \left(\tan ^{-1} x\right)

View answer and explanation

Correct option: C

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

Option C

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

is correct because it is the differential equation obtained by differentiating

y=\sec \left(\tan ^{-1} x\right)

and eliminating the trigonometric functions.

368 Medium

368 of 975

Application +4 | -1

By a suitable substitution, the equation

y^3 \frac{d y}{d x}+x+y^2=0

can be transformed to

View answer and explanation

Correct option: B

Homogeneous

Option B (Homogeneous) is correct because the given equation can be transformed into a homogeneous differential equation by using the substitution

z = x + y^2

369 Easy

369 of 975

Application +4 | -1

The differential equation of the rectangular hyperbola whose axes are the asymptotes of the hyperbola, is

View answer and explanation

Correct option: B

x\frac{{dy}}{{dx}} = - y

Option B (

x\frac{{dy}}{{dx}} = - y

) is correct because it represents the differential equation of a rectangular hyperbola whose axes are the asymptotes.

370 Medium

370 of 975

Application +4 | -1

The general solution of

ydx - xdy - 3{x^2}{y^2}{e^{{x^3}}}dx = 0

, is equal to

View answer and explanation

Correct option: A

\frac{x}{y} = {e^{{x^3}}} + C

Option A (

\frac{x}{y} = {e^{{x^3}}} + C

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

371 Easy

371 of 975

Application +4 | -1

If integrating factor of

x\left( {1 - {x^2}} \right)dy + \left( {2{x^2}y - y - a{x^3}} \right)dx = 0

{e^{\int Pdx}}

, then

P

is equal to

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it represents the coefficient of

y

when the given differential equation is written in the standard form

\frac{{dy}}{{dx}} + Py = Q

372 Medium

372 of 975

Application +4 | -1

The solution curve, of the differential equation

2 {y} \frac{{dy}}{{dx}}+3=5 \frac{{dy}}{{dx}}

, passing through the point

(0,1)

is a conic, whose vertex lies on the line:

View answer and explanation

Correct option: A

2 x+3 y=9

Option A (

2 x+3 y=9

) is correct because the vertex of the conic (parabola) lies on this line, after solving the differential equation and applying the given point.

373 Easy

373 of 975

Application +4 | -1

The differential equation of all ellipses with centres at the origin and the ends of one axis of symmetry at

(\pm 1,0)

, is

View answer and explanation

Correct option: A

\left(x^2-1\right) y^{\prime}-x y=0

Option A

\left(x^2-1\right) y^{\prime}-x y=0

is correct. It is derived by eliminating the arbitrary constant from the equation of the ellipse.

374 Easy

374 of 975

Application +4 | -1

If integrating factor of

x\left( {1 - {x^2}} \right)dy + \left( {2{x^2}y - y - a{x^3}} \right)dx = 0\mathrm{\,\, is \,\,}{e^{\int P dx}}

then

P

is equal to

View answer and explanation

Correct option: D

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

Option D (

\frac{2 x^2-1}{x(1-x^2)}

) is correct because it represents the coefficient of 'y' in the standard form of the given linear differential equation, which is required to find the integrating factor.

375 Easy

375 of 975

Analyzing +4 | -1

The slope of the tangent to a curve

C: y=y(x)

at any point

(x, y)

on it is

\frac{2 e^{2 x}-6 e^{-x}+9}{2+9 e^{-2 x}}

. If

C

passes through the points

\left(0, \frac{1}{2}+\frac{\pi}{2 \sqrt{2}}\right)

and

\left(\alpha, \frac{1}{2} e^{2 \alpha}\right)

, then

e^\alpha

is equal to

View answer and explanation

Correct option: B

\frac{3}{\sqrt{2}}\left(\frac{3+\sqrt{2}}{3-\sqrt{2}}\right)

Option B is correct. The value of

e^\alpha

is found by solving the differential equation and using the given points to find the constant of integration and then solving for

e^\alpha

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376 Medium

376 of 975

Application +4 | -1

The equation of curve through point (1,0) which satisfies the differential equation

\left( {1 + {y^2}} \right)dx - xy\mathrm{ }dy = 0

View answer and explanation

Correct option: B

{x^2} - {y^2} = 1

Option B (

{x^2} - {y^2} = 1

) is correct because it is the solution to the differential equation that passes through the point (1,0).

377 Medium

377 of 975

Application +4 | -1

General solution of the equation

\frac{d y}{d x}+\frac{1}{x}=\frac{e^y}{x^2}

View answer and explanation

Correct option: A

2 x=e^y\left(2 c x^2+1\right)

Option A (

2 x=e^y(2 c x^2+1)

) is the correct general solution obtained by solving the differential equation.

378 Easy

378 of 975

Understanding +4 | -1

The order and degree of the differential equation

5{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)^5} + 4{\left( {\frac{{{d^3}y}}{{d{x^3}}}} \right)^2} + {\left( {\frac{{dy}}{{dx}}} \right)^3} + 2y + {x^3} = 0

are respectively

View answer and explanation

Correct option: B

\left( {3,2} \right)

Option B (

\left( {3,2} \right)

) is correct because the highest order derivative in the equation is

\frac{{{d^3}y}}{{d{x^3}}}

(order 3) and its power is 2 (degree 2).

379 Medium

379 of 975

Application +4 | -1

The solution of

{e^{\frac{{dy}}{{dx}}}} = \left( {x + 1} \right),y\left( 0 \right) = 3

View answer and explanation

Correct option: B

y = \left( {x + 1} \right)\log \left| {x + 1} \right| - x + 3

Option B,

y = \left( {x + 1} \right)\log \left| {x + 1} \right| - x + 3

is the correct solution obtained by integrating the given differential equation and applying the initial condition.

380 Medium

380 of 975

Application +4 | -1

The solution of

(y+x+5) \mathrm{d} y=(y-x+

\mathrm{d} x

View answer and explanation

Correct option: C

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

The correct answer is C (

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

) because it is the solution to the homogeneous differential equation after substitution and integration.

381 Easy

381 of 975

Application +4 | -1

If the function

y=\log \frac{1}{1+x}

satisfies the relation

x y^{\prime}

+1=f(y)

then

f(y)

is equal to

View answer and explanation

Correct option: C

e^y

Option D (

e^{-y}

) is correct because it satisfies the given relation after substituting the derivative of the function.

382 Easy

382 of 975

Application +4 | -1

If the solution of

\frac{d y}{d x}=\frac{a x+3}{2 y+f}

represents a circle, then the value of

a

View answer and explanation

Correct option: A

Option A (2) is correct because for the given differential equation to represent a circle, the coefficient of x² and y² must be equal, leading to a = -2.

383 Easy

383 of 975

Application +4 | -1

The order and degree respectively of the differential equation of all tangent lines to parabola

x^2=

2 y

View answer and explanation

Correct option: A

1,2

Option A (1,2) is correct because the differential equation representing the tangent lines to the parabola has an order of 1 and a degree of 2.

384 Hard

384 of 975

Analyzing +4 | -1

y=(1+x)\left(x^{2}+1\right)\left(x^{4}+1\right)\left(x^{8}+1\right)\left(x^{16}+1\right)

, then find the value of

y^{\prime \prime}-y^{\prime}

x=-1

View answer and explanation

Correct option: C

-496

Option C (-496) is correct because after simplifying the expression for y and differentiating twice, substituting x = -1 yields y'' - y' = -496.

385 Easy

385 of 975

Application +4 | -1

If the differential equation representing the family of all circles touching

y

-axis at the origin is

x^2-y^2

=f(x) y \frac{d x}{d y}

then

f(x)

is equal to

View answer and explanation

Correct option: A

2 x

Option A (

2x

) is correct because it is the function

f(x)

that satisfies the differential equation representing the family of circles touching the y-axis at the origin.

386 Medium

386 of 975

Application +4 | -1

y_1(x)

is a solution of the differential equation

\frac{d y}{d x}-f(x) y=0

, then a solution of the differential equation

\frac{d y}{d x}+f(x) y=r(x)

can be

y=

View answer and explanation

Correct option: A

\frac{1}{y_1(x)} \int r(x) y_1(x) d x

Option A (

\frac{{1}}{{y_1(x)}} \int r(x) y_1(x) dx

) is correct because it represents the solution obtained using the integrating factor method for a first-order linear differential equation.

387 Easy

387 of 975

Understanding +4 | -1

Solution of the differential equation

xdy - ydx = 0

represents a

View answer and explanation

Correct option: D

Straight line

Option D (Straight line) is correct because the solution to the differential equation

xdy - ydx = 0

y = cx

, which represents a straight line passing through the origin.

388 Medium

388 of 975

Application +4 | -1

The differential equation

\frac{d y}{d x}=\frac{\sqrt{1-y^2}}{y}

determines a family of circles with

View answer and explanation

Correct option: C

fixed radius 1 and variable centres along the

x

-axis

Option C, fixed radius 1 and variable centres along the

x

-axis, is correct because the differential equation represents a family of circles with this property.

389 Easy

389 of 975

Application +4 | -1

x\frac{{dy}}{{dx}} = y\left( {\log y - \log x + 1} \right)

, then the solution of the equation is

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the solution obtained by solving the given homogeneous differential equation.

390 Hard

390 of 975

Application +4 | -1

The solution of the differential equation

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

View answer and explanation

Correct option: A

a y^2=e^{x^2 / y^2}

Option A (

a y^2=e^{x^2 / y^2}

) is the correct solution to the given differential equation, which is solved by substituting

y=vx

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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