Let y = y(x) be the solution of the differential equation ${e^x}\sqrt {1 - {\rm{ }}{y^2}} {\rm{ }}dx{\rm{ }} + \left( {\frac{y}{x}} \right)dy\, = {\rm{ }}0,{\rm{ }}y\left( 1 \right){\rm{ }} = {\rm{ }} - 1.$ Then, the value of ${\left[ {y\left( 3 \right)} \right]^2}$ is equal to

${\rm{1}} - {\rm{ 4}}{{\rm{e}}^6}$

${\rm{1}} - {\rm{ 4}}{{\rm{e}}^3}$

Let $y = y(x)$ satisfies the equation $\frac{{dy}}{{dx}} - \left| A \right| = 0$, for all $x > 0$ where $A = \left[ {\begin{array}{*{20}{c}} y&{\sin x}&1\\ 0&{ - 1}&1\\ 2&0&{\frac{1}{x}} \end{array}} \right]$. If $y(\pi ) = \pi + 2$, then the value of $y\left( {\frac{\pi }{2}} \right)$ is

$\frac{\pi }{2} - \frac{1}{\pi }$

$\frac{{3\pi }}{2} - \frac{1}{\pi }$

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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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301 Medium

301 of 975

Application +4 | -1

A function

y=f(x)

satisfies

(x+1) \cdot f^{\prime}(x)-2\left(x^2+x\right) f(x)=\frac{e^{x^2}}{(x+1)}, \forall x>-1

.
If

f(0)=5

, then

f(x)

View answer and explanation

Correct option: B

\left(\frac{6 x+5}{x+1}\right) \cdot e^{x^2}

Option B (

\left(\frac{{6 x+5}}{{x+1}}\right) \cdot e^{x^2}

) is correct because it satisfies the given differential equation and the initial condition f(0)=5.

302 Medium

302 of 975

Application +4 | -1

The solution of the differential equation

\frac{{dy}}{{dx}} = \frac{{x\log {x^2} + x}}{{\sin y + y\cos y}}

, is

View answer and explanation

Correct option: A

y\sin y = {x^2}\log x + C

Option A (

y\sin y = {x^2}\log x + C

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

303 Easy

303 of 975

Application +4 | -1

The curve satisfying

y \mathrm{~d} x-x \mathrm{~d} y+\log x \mathrm{~d} x=0

(x > 0)

passing through

(1,-1)

View answer and explanation

Correct option: A

y+\log x+1=0

Option A (

y+\log x+1=0

) is correct because it satisfies the given differential equation and passes through the point (1, -1).

304 Medium

304 of 975

Analyzing +4 | -1

The differential equation whose solution is

{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {a^2}

is (

a

is a constant)

View answer and explanation

Correct option: B

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

Option B (

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

) is correct because it is the differential equation obtained by differentiating the given equation twice and eliminating the arbitrary constants h and k.

305 Easy

305 of 975

Application +4 | -1

The solution of differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{y}{x}+2 \frac{\phi(y / x)}{\phi^{\prime}(y / x)}

View answer and explanation

Correct option: C

\phi(y / x)=k x^2

Option C (

\phi(y / x)=k x^2

) is correct because it is the solution to the given differential equation after applying the substitution

y/x = u

306 Easy

306 of 975

Application +4 | -1

The solution of the differential equation

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

View answer and explanation

Correct option: B

\log _e|x+y|+\frac{2 x y}{(x+y)^2}=0

Option B (

\log _e|x+y|+\frac{{2 x y}}{{(x+y)^2}}=0

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

y=vx

and applying the initial condition

y(1)=0

307 Easy

307 of 975

Application +4 | -1

Solution of differential equation

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

View answer and explanation

Correct option: D

x^2+y^2=p x^3

(where

p

is a constant)

Option D (

x^2+y^2=p x^3

) is the correct solution to the homogeneous differential equation, obtained by substituting

y=vx

and integrating.

308 Easy

308 of 975

Application +4 | -1

The differential equation of the family of curves

y=e^x(A \cos x+B \sin x)

View answer and explanation

Correct option: B

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

Option B (

\frac{{d^2y}}{{dx^2}} - 2\frac{{dy}}{{dx}} + 2y = 0

) is correct because it is the differential equation of the given family of curves obtained by differentiating twice and eliminating the arbitrary constants A and B.

309 Easy

309 of 975

Application +4 | -1

Solution of the equation

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

View answer and explanation

Correct option: A

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

Option A (

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

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

y=vx

310 Easy

310 of 975

Application +4 | -1

Solution of the differential equation

2 y \sin x \frac{\mathrm{d} y}{\mathrm{~d} x}=2 \sin x \cos x-y^2 \cos x

satisfying

y(\pi / 2)=1

is given by

View answer and explanation

Correct option: A

y^2=\sin x

Option A (

y^2=\sin x

) is correct because it is the solution obtained by solving the given differential equation and applying the initial condition y(

\pi / 2

)=1.

311 Medium

311 of 975

Application +4 | -1

The solution of the differential equation

\left(1-x y-x^5 y^5\right) d x-x^2\left(x^4 y^4+1\right) d y=0

is given by

View answer and explanation

Correct option: A

x=c e^{x y+\frac{1}{5} x^5 y^5}

Option A (

x=c e^{x y+\frac{1}{5} x^5 y^5}

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

312 Medium

312 of 975

Application +4 | -1

Solution of the differential equation

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

View answer and explanation

Correct option: B

x y \sqrt{y^2-x^2}=c

Option B (

x y \sqrt{y^2-x^2}=c

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

y=vx

313 Easy

313 of 975

Application +4 | -1

The solution of the differential equation

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

View answer and explanation

Correct option: B

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

Option B (

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

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

314 Easy

314 of 975

Analyzing +4 | -1

y=(A+B x) e^{m x}+(m-1)^{-2} e^x

, then

\frac{d^2 y}{d x^2}-2 m \frac{d y}{d x}+m^2 y

is equal to

View answer and explanation

Correct option: A

e^x

Option A (

e^x

) is correct because substituting the given

y

into the differential equation results in

e^x

315 Medium

315 of 975

Application +4 | -1

Let

y=y(x)

be the solution of the differential equation

\frac{d y}{d x}+2 y=f(x)

, where

f(x)= \begin{cases}1, & x \in[0,1] \\ 0, & \text { otherwise }\end{cases}

y(0)=0

, then

y\left(\frac{3}{2}\right)

View answer and explanation

Correct option: A

\frac{e^2-1}{2 e^3}

Option A ((e^2-1)/(2e³)) is correct because solving the differential equation with the piecewise function and applying the initial condition yields y(3/2) = (e^2-1)/(2e³).

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316 Medium

316 of 975

Application +4 | -1

x d y=\left(2 y+2 x^4+x^2\right) d x, y(1)=0

then

y(e)=

View answer and explanation

Correct option: D

\mathrm{e}^4

Option D (

e^4

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

317 Easy

317 of 975

Application +4 | -1

Let y = y(x) be the solution of the differential equation

{e^x}\sqrt {1 - \mathrm{ }{y^2}} \mathrm{ }dx\mathrm{ } + \left( {\frac{y}{x}} \right)dy\, = \mathrm{ }0,\mathrm{ }y\left( 1 \right)\mathrm{ } = \mathrm{ } - 1.

Then, the value of

{\left[ {y\left( 3 \right)} \right]^2}

is equal to

View answer and explanation

Correct option: B

\mathrm{1} - \mathrm{ 4}{\mathrm{e}^6}

Option B (

\mathrm{1} - \mathrm{ 4}{\mathrm{e}^6}

) is correct because solving the differential equation with the given initial condition and then evaluating at x=3 leads to this result.

318 Easy

318 of 975

Application +4 | -1

If the solution of the differential equation

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

satisfies

y\left( 0 \right) = 0,

then the value of

y\left( 2 \right)

is_____.

View answer and explanation

Correct option: C

Option C (0) is correct because after solving the differential equation and applying the initial condition, the value of y(2) is found to be 0.

319 Medium

319 of 975

Application +4 | -1

Solution of differential equations

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

View answer and explanation

Correct option: C

\left|\frac{\sin x}{x y}\right|=c

Option C (

\left|\frac{{\sin x}}{{x y}}\right|=c

) is correct because it represents the solution obtained after integrating the given differential equation and simplifying the result.

320 Medium

320 of 975

Application +4 | -1

Solution of equation

\frac{d y}{d x}=\frac{3 x-4 y-2}{3 x-4 y-3}

View answer and explanation

Correct option: B

x-y+c=\log (3 x-4 y+1)

Option B (

x-y+c=\log (3 x-4 y+1)

) is correct because it is the solution obtained by substituting

3x-4y=z

and integrating the resulting differential equation.

321 Medium

321 of 975

Application +4 | -1

The solution of differential equation

x \,d x+y \,d y=a\left(x^2+y^2\right) d y

, is

View answer and explanation

Correct option: B

x^2+y^2=C e^{2 a y}

Option B (

x^2+y^2=C e^{2 a y}

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

322 Medium

322 of 975

Analyzing +4 | -1

Solution to the differential equation

\frac{x+\frac{x^3}{3 !}+\frac{x^3}{5 !}+\cdots}{1+\frac{x^2}{2 !}+\frac{x^4}{4 !}+\cdots}=\frac{d x-d y}{d x+d y}

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it is the solution obtained by solving the given differential equation using componendo and dividendo and integrating.

323 Easy

323 of 975

Application +4 | -1

The solution of the differential equation

1-\frac{d y}{d x}=e^{x-y}

View answer and explanation

Correct option: A

(x+C) e^{x-y}+1=0

Option A (

(x+C) e^{x-y}+1=0

) is correct because solving the separable differential equation yields this solution.

324 Easy

324 of 975

Application +4 | -1

Let

y = y(x)

satisfies the equation

\frac{{dy}}{{dx}} - \left| A \right| = 0

, for all

x > 0

where

A = \left[ {\begin{array}{cccccccccccccccccccc} y&{\sin x}&1\\ 0&{ - 1}&1\\ 2&0&{\frac{1}{x}} \end{array}} \right]

. If

y(\pi ) = \pi + 2

, then the value of

y\left( {\frac{\pi }{2}} \right)

View answer and explanation

Correct option: A

\frac{\pi }{2} + \frac{4}{\pi }

Option A (

\frac{\pi}{2} + \frac{4}{\pi}

) is correct because solving the differential equation and applying the given condition yields this result.

325 Medium

325 of 975

Application +4 | -1

The solution of the differential equation

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

View answer and explanation

Correct option: B

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

Option B (

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

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

326 Medium

326 of 975

Application +4 | -1

The solution curve of the differential equation

y \frac{d x}{d y}=x\left(\log _e x-\log _e y+1\right), x>0, y>0 \text { passing }

through the point

(e, 1)

View answer and explanation

Correct option: A

\left|\log _e \frac{x}{y}\right|=y

Option A (

\left|\log _e \frac{x}{y}\right|=y

) is correct because it involves solving the given differential equation and applying the initial condition to find the particular solution.

327 Easy

327 of 975

Application +4 | -1

The solution of

\frac{{dy}}{{dx}} = \frac{{ax + h}}{{by + k}}

represents a parabola, when

View answer and explanation

Correct option: C

a = 0,b \ne 0

Option C (

a = 0,b \ne 0

) is correct because when a=0, the equation represents a parabola.

328 Easy

328 of 975

Application +4 | -1

Solution of the equation

x{\left( {\frac{{dy}}{{dx}}} \right)^2} + 2\sqrt {xy} \frac{{dy}}{{dx}} + y = 0

View answer and explanation

Correct option: D

\sqrt x + \sqrt y = c

Option D (

\sqrt x + \sqrt y = c

) is correct because it is the solution obtained by rewriting the differential equation as a perfect square and integrating.

329 Easy

329 of 975

Application +4 | -1

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

View answer and explanation

Correct option: B

-\left(\frac{d^2 y}{d x^2}\right)\left(\frac{d y}{d x}\right)^{-3}

Option B is correct because it shows the correct application of the chain rule and inverse function differentiation to find the second derivative.

330 Easy

330 of 975

Application +4 | -1

An integrating factor of the differential equation

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

View answer and explanation

Correct option: B

x

Option B (

x

) is correct because it is the integrating factor that makes the given differential equation exact.

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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