The solution of $\left( {1 + {x^2}} \right)\frac{{dy}}{{dx}} + 2xy - 4{x^2} = 0$ is

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

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

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

The solution of $\frac{{dy}}{{dx}} + 1 = {\rm{cosec}}\left( {x + y} \right)$ is

$\cos \left( {c + y} \right) = c$

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

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

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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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Showing 391-420 of 975 questions

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391 Easy

391 of 975

Application +4 | -1

Let the solution curve

y=y(x)

of the differential equation

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

pass through the point

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

. Then,

\lim\limits_{x \to \infty } {e^x}y(x)

is equal to

View answer and explanation

Correct option: B

\frac{3 \pi}{4}

Option B (

\frac{{3 \pi}}{{4}}

) is correct because after solving the differential equation and applying the given conditions, the limit evaluates to this value.

392 Easy

392 of 975

Application +4 | -1

The solution of

y^{\prime}+x y=x y^{-1}, y(0)=2

is given by

View answer and explanation

Correct option: C

y^2=3+e^{-x^2}

Option C (

y^2=3+e^{-x^2}

) is correct because it is the solution to the differential equation

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

with the initial condition

y(0)=2

393 Easy

393 of 975

Application +4 | -1

\cos x\frac{{dy}}{{dx}} - y\sin x = 6x,\left( {0 < x < \frac{\pi }{2}} \right)\,\,{\text{and}}\,\,y\left( {\frac{\pi }{3}} \right) = 0,{\text{then}}\,\,y\left( {\frac{\pi }{6}} \right)

is equal to:

View answer and explanation

Correct option: C

-\frac{\pi^2}{2 \sqrt{3}}

Option C (

-\frac{\pi^2}{2 \sqrt{3}}

) is correct because it is the value of y(

\frac{\pi}{6}

) obtained after solving the given differential equation and applying the initial condition y(

\frac{\pi}{3}

) = 0.

394 Easy

394 of 975

Application +4 | -1

The solution of

\left( {1 + {x^2}} \right)\frac{{dy}}{{dx}} + 2xy - 4{x^2} = 0

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the solution obtained by solving the linear differential equation using an integrating factor.

395 Medium

395 of 975

Application +4 | -1

The solution of

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

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is the solution obtained by substituting

x+y=t

and integrating the resulting equation.

396 Easy

396 of 975

Application +4 | -1

Solution of

x\frac{{dy}}{{dx}} + y = x{e^x}

, is

View answer and explanation

Correct option: B

xy = {e^x}\left( {x - 1} \right) + C

Option B (

xy = {e^x}\left( {x - 1} \right) + C

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

397 Medium

397 of 975

Application +4 | -1

A curve passes through the point

\left(1, \frac{\pi}{4}\right)

and its slope at any point is given by

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

. Then, the curve has the equation

View answer and explanation

Correct option: A

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

Option A,

y=x \tan ^{-1}(\ln \frac{e}{x})

, is the correct equation of the curve that satisfies the given differential equation and passes through the point

(1, \frac{\pi}{4})

398 Easy

398 of 975

Application +4 | -1

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

and

y=1

when

x=1

, then

y=2

when

5 x=\ldots (x \neq 0)

View answer and explanation

Correct option: A

Option A (8) is correct because solving the differential equation and using the given condition, we find that 5x = 8 when y = 2.

399 Medium

399 of 975

Application +4 | -1

If the algebraic sub-tangent at any point of a curve is equal to half of the sum of the coordinates of the point, then the equation of the curve is

View answer and explanation

Correct option: B

(x-y)^2=c y

Option C (

(x-y)^2=c x

) is the correct answer because it satisfies the given condition about the algebraic sub-tangent after solving the differential equation.

400 Easy

400 of 975

Understanding +4 | -1

The order and degree of the differential equation

\frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^{1 / 3}+x^{1 / 4}=0

View answer and explanation

Correct option: B

order -2 , degree -3

Option B (order -2 , degree -3) is correct because the order is the highest derivative and the degree is the power of the highest derivative after removing radicals.

401 Easy

401 of 975

Understanding +4 | -1

The order of the differential equation

\frac{{{d^2}y}}{{d{x^2}}} = \sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^3}}

, is

View answer and explanation

Correct option: A

Option A (2) is correct because the highest order derivative in the equation is the second derivative, making the order 2.

402 Easy

402 of 975

Application +4 | -1

The differential equation of the family of curves represented by the equation

x^2 y=a

, is

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because differentiating

x^2y=a

and eliminating 'a' leads to this differential equation.

403 Medium

403 of 975

Analyzing +4 | -1

f(x)

is a differentiable function in the interval

((0, \infty)

such that

f(a)=1

and

\lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1

, for each

x > 0

, then

\mathrm{f}\left(\frac{3}{2}\right)

is equal to:

View answer and explanation

Correct option: D

\frac{31}{18}

Option D (

\frac{{31}}{{18}}

) is correct because it is the value of f(3/2) obtained after solving the given differential equation and using the initial condition f(a) = 1.

404 Medium

404 of 975

Analyzing +4 | -1

The differential equation of the family of curves

y = {e^{2x}}\left( {a\cos x + b\sin x} \right)

, where

a

and

b

are arbitrary constants, is given by

View answer and explanation

Correct option: A

{y_2} - 4{y_1} + 5y = 0

Option A (

{y_2} - 4{y_1} + 5y = 0

) is correct because it represents the differential equation obtained after differentiating the given equation twice and eliminating the arbitrary constants a and b.

405 Medium

405 of 975

Application +4 | -1

Solution of

\cos x \frac{d y}{d x}+y \sin x=1

View answer and explanation

Correct option: B

y \sec x \tan x=c

The correct answer is C (

y \tan x=\sec x+c

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

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406 Easy

406 of 975

Application +4 | -1

e^y+x y=e

, then the ordered pair

\left(\frac{d y}{d x}, \frac{d^2 y}{d x^2}\right)

x=0

is equal to

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the ordered pair of the first and second derivatives evaluated at x=0.

407 Easy

407 of 975

Application +4 | -1

\frac{d y}{d x}=y+3 > 0

and

y(0)=2

, then

y(\ln 2)

is equal to:

View answer and explanation

Correct option: D

Option D (7) is correct because solving the differential equation and applying the initial condition yields y(\ln 2) = 7.

408 Medium

408 of 975

Application +4 | -1

The equation of family of a curve is

{y^2} = 4a\left( {x + a} \right)

then differential equation of the family is

View answer and explanation

Correct option: C

y = 2y'x + y{y'^2}

Option C (

y = 2y'x + y{y'^2}

) is the correct differential equation obtained by differentiating the given equation and eliminating the arbitrary constant 'a'.

409 Easy

409 of 975

Application +4 | -1

Solution of the differential equation

x \mathrm{~d} y-y \mathrm{~d} x-\sqrt{x^2+y^2} \mathrm{~d} x=0

View answer and explanation

Correct option: B

y+\sqrt{x^2+y^2}=C x^2

Option B (

y+\sqrt{x^2+y^2}=C x^2

) is correct because it is the solution obtained by solving the homogeneous differential equation using the substitution y=vx.

410 Medium

410 of 975

Application +4 | -1

Let

x=x(t)

and

y=y(t)

be solutions of the differential equations

\frac{d x}{d t}+a x=0

and

\frac{d y}{d t}+b y=0

respectively,

a, b \in \mathbb{R}.

Given that

x(0)=2 ; y(0)=1

and

3 y(1)=2 x(1),

the value of

t,

for which

x(t)=y(t),

is:

View answer and explanation

Correct option: C

\log _{\frac{4}{3}} 2

Option C is correct because it involves solving two first-order differential equations and then using the given condition 3y(1) = 2x(1) to find the value of t for which x(t) = y(t).

411 Easy

411 of 975

Application +4 | -1

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

satisfies the relation

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

=k

then the value of

k

is1

View answer and explanation

Correct option: A

Option A (1) is correct because it is the value of k that satisfies the given relation after substituting the derivative of the function.

412 Medium

412 of 975

Application +4 | -1

y=a x^{n+1}+b x^{-n}

, then

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

equals to

View answer and explanation

Correct option: B

n(n+1) y

Option B (

n(n+1) y

) is correct because after differentiating twice and multiplying by

x^2

, the result is

n(n+1)

times the original function

y

413 Medium

413 of 975

Analyzing +4 | -1

The differential equation of the family of parabolas with focus at the origin and the

x

-axis as axis, is

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it represents the differential equation of the family of parabolas with focus at the origin and the x-axis as the axis.

414 Easy

414 of 975

Application +4 | -1

The differential equation

\left( {{e^x} + 1} \right)y\mathrm{ }dy = \left( {y + 1} \right){e^x}dx

, has the solution

View answer and explanation

Correct option: D

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

Option D (

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

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

415 Medium

415 of 975

Application +4 | -1

The solution of the differential equation

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

is
(here

c

is an arbitrary constant)

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the solution to the given homogeneous differential equation.

416 Medium

416 of 975

Application +4 | -1

Solution of the differential equation

\left(x \sin ^2\left(\frac{y}{x}\right)-y\right) d x+x d y=0, y=\frac{\pi}{4}

when

x=1

View answer and explanation

Correct option: A

\cot \left(\frac{y}{x}\right)=\log (\operatorname{ex})

Option A,

\cot \left(\frac{y}{x}\right)=\log (\operatorname{ex})

, is the correct solution obtained by solving the homogeneous differential equation and applying the given initial condition.

417 Medium

417 of 975

Application +4 | -1

Solution of

\frac{d y}{d x}=\frac{y}{x}+\tan \frac{y}{x}

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is the solution obtained by substituting

y=vx

and integrating the resulting separable differential equation.

418 Medium

418 of 975

Application +4 | -1

The solution of

x^3 \frac{\mathrm{d} y}{\mathrm{~d} x}+4 x^2

tan

y=

e^x \sec y

satisfying

y(1)=0

View answer and explanation

Correct option: B

\sin y=e^x(x-1) x^{-4}

Option B (

\sin y=e^x(x-1) x^{-4}

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

419 Easy

419 of 975

Application +4 | -1

The solution of

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

View answer and explanation

Correct option: B

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

Option B (

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

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

420 Medium

420 of 975

Application +4 | -1

The solution of

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

View answer and explanation

Correct option: B

\log x+\frac{y^3}{x}=k

Option B (

\log x+\frac{y^3}{x}=k

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

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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