Let $y = y\left( x \right)$ be the solution of the differential equation $\sin x\frac{{dy}}{{dx}} + y\cos x = 4x,x \in \left( {0,\pi } \right)$ If $y\left( {\frac{\pi }{2}} \right) = 0$, then $y\left( {\frac{\pi }{6}} \right)$ is equal to

$\frac{4}{{9\sqrt 3 }}{\pi ^2}$

$\frac{{ - 8}}{{9\sqrt 3 }}{\pi ^2}$

Solution of the equation $\frac{d y}{d x}=\sqrt{y-x}$ is

$2 \sqrt{y-x}+\log |\sqrt{y-x}-1|=x+c$

$2 \sqrt{y-x}+\log |y-x-1|=x+c$

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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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331 Medium

331 of 452

Application +4 | -1

The solution of the equation

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

View answer and explanation

Correct option: A

(x-y)(x+2 y)^2=c

Option A (

(x-y)(x+2 y)^2=c

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

y=vx

332 Medium

332 of 452

Application +4 | -1

Let

y=y(x)

be the solution of the differential equation

\frac{d y}{d x}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x \tan x)}, x \in\left(0, \frac{\pi}{2}\right)

satisfying the condition

y\left(\frac{\pi}{4}\right)=2

. Then,

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

View answer and explanation

Correct option: D

\sqrt{3}\left(2+\log _e \sqrt{3}\right)

Option D (

\sqrt{3}\left(2+\log _e \sqrt{3}\right)

) is correct because it is the value of y(

\frac{\pi}{3}

) obtained by solving the given differential equation with the initial condition y(

\frac{\pi}{4}

)=2.

333 Medium

333 of 452

Application +4 | -1

The solution of the differential equation

\frac{{dy}}{{dx}} = \frac{1}{{{x^2} + {y^2}}}

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because it's the solution obtained by solving the differential equation after rewriting it in the form

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

and finding the integrating factor.

334 Medium

334 of 452

Application +4 | -1

The solution of

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

is given by

View answer and explanation

Correct option: B

x^2-x y+y^2+x-y=C

The correct answer is C (

x^2-x y+y^2-x+y=C

) because it is the general solution to the given differential equation after reducing it to a homogeneous equation.

335 Medium

335 of 452

Application +4 | -1

A particles moves in a straight line with a velocity given by

\frac{{dx}}{{dt}} = x + 1

(

x

is the distance described). The time taken by a particle to traverse a distance of 99 metres is

View answer and explanation

Correct option: B

2{\log _e}10

Option B (

2{\log _e}10

) is correct because it is the time calculated by solving the differential equation and applying the given conditions.

336 Medium

336 of 452

Application +4 | -1

The general solution of

x \frac{d y}{d x}+(\log x) y=x^{1-\frac{1}{2} \log x}

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it represents the general solution of the given differential equation after finding the integrating factor and integrating.

337 Medium

337 of 452

Application +4 | -1

The solution of

\frac{d y}{d x}+y=e^{-x}, y(0)=0

, is

View answer and explanation

Correct option: D

y=x e^{-x}

Option D (

y=x e^{-x}

) is correct because it is the solution obtained by solving the given differential equation with the given initial condition.

338 Medium

338 of 452

Application +4 | -1

The differential equation of family of circles with fixed radius 5 units and centre on the line

y=2

, is

View answer and explanation

Correct option: A

(y-2)^2 y^{\prime 2}=25-(y-2)^2

Option D (

(y-2) y^{\prime 2}=25-(y-2)^2

) is incorrect. The correct answer is A (

(y-2)^2 y^{\prime 2}=25-(y-2)^2

) because it represents the differential equation of the family of circles with the given conditions.

339 Medium

339 of 452

Understanding +4 | -1

The order and degree of the differential equation

\frac{{{d^2}y}}{{d{x^2}}} = \sqrt[3]{{1 - {{\left( {\frac{{dy}}{{dx}}} \right)}^4}}}

are respectively

View answer and explanation

Correct option: A

2,3

Option A (2,3) is correct because after cubing the equation to remove the cube root, the highest order derivative is 2 and its power is 3.

340 Medium

340 of 452

Analyzing +4 | -1

Consider the differential equation

e^x(y d x-d y)=e^{-x}(y d x+d y)

. Let

{y}={f}({x})

be the particular solution of the differential equation which passes through

(0,2)

. Then range of

g(x)=\log _2(f(x))

, is

View answer and explanation

Correct option: B

[1, \infty)

Option B ([1, ∞)) is correct because solving the differential equation and applying the initial condition leads to f(x) = e^x + e^(-x), and the range of log_2(f(x)) is [1, ∞).

341 Medium

341 of 452

Application +4 | -1

The solution of the differential equation

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

satisfying the condition

y\left( 1 \right) = 1

View answer and explanation

Correct option: A

y = x\log x + x

Option A (

y = x\log x + x

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

342 Medium

342 of 452

Application +4 | -1

The equation of one of the curves whose slope at any point is equal to

y + 2x

View answer and explanation

Correct option: B

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

Option B (

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

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

343 Medium

343 of 452

Analyzing +4 | -1

Let

F

denotes the family of ellipses whose centre is at the origin and major axis is the

y

-axis. Then, equation of the family

F

View answer and explanation

Correct option: C

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

Option C is correct because the differential equation is derived by differentiating the equation of the ellipse and eliminating the arbitrary constants.

344 Medium

344 of 452

Application +4 | -1

The differential equation for the family of curves

{x^2} + {y^2} - 2ay = 0

, where

a

is an arbitrary constant, is

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it is the differential equation obtained by eliminating the arbitrary constant 'a' from the given family of curves.

345 Medium

345 of 452

Application +4 | -1

Let

y = y\left( x \right)

be the solution of the differential equation

\sin x\frac{{dy}}{{dx}} + y\cos x = 4x,x \in \left( {0,\pi } \right)

y\left( {\frac{\pi }{2}} \right) = 0

, then

y\left( {\frac{\pi }{6}} \right)

is equal to

View answer and explanation

Correct option: C

- \frac{8}{9}{\pi ^2}

Option C (

- \frac{8}{9}{\pi ^2}

) is correct because it is the value of

y(\frac{\pi}{6})

obtained by solving the given differential equation and applying the initial condition.

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346 Medium

346 of 452

Application +4 | -1

The degree of the differential equation satisfied by the curve

\sqrt{1+\mathrm{x}}-\mathrm{a} \sqrt{1+\mathrm{y}}=1

, is

View answer and explanation

Correct option: A

Option A (1) is correct because after differentiating and eliminating the constant, the highest order derivative has a power of 1.

347 Medium

347 of 452

Application +4 | -1

Solution of the equation

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

View answer and explanation

Correct option: A

2 \sqrt{y-x}+2 \log |\sqrt{y-x}-1|=x+c

Option A (

2 \sqrt{y-x}+2 \log |\sqrt{y-x}-1|=x+c

) is correct because it is the solution obtained after using the substitution

y-x = z^2

and integrating the resulting equation.

348 Medium

348 of 452

Application +4 | -1

The solution

y(x)

of the differential equation

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

when

y_1(0)=1

and

y(0)=0

View answer and explanation

Correct option: A

-\frac{\sin 3 x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3} x-1

Option A (

-\frac{\sin 3 x}{9}+e^x+\frac{x^4}{12}+\frac{1}{3} x-1

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

349 Medium

349 of 452

Application +4 | -1

The solution of

\frac{d y}{d x}+2 y \tan x=\sin x

, is

View answer and explanation

Correct option: B

y \sec ^2 x=\sec x+c

Option B (

y \sec ^2 x=\sec x+c

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

350 Medium

350 of 452

Application +4 | -1

General solution of the equation

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

View answer and explanation

Correct option: B

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

Option D (

2 y=e^{\operatorname{{Tan}}^{-1} x}+c e^{-\operatorname{{Tan}}^{-1}} x

) is incorrect. The correct answer is C (

2 y=e^{\operatorname{{Tan}}^{-1} x}+c e^{-\operatorname{{Tan}}^{-1}} x

). The error likely stems from incorrect handling of the integrating factor or the constant of integration.

351 Medium

351 of 452

Understanding +4 | -1

The differential equation of all circles in the first quadrant which touch the coordinate axes is of order

View answer and explanation

Correct option: A

Option A (1) is correct because the family of circles is defined by one parameter, leading to a first-order differential equation.

352 Medium

352 of 452

Application +4 | -1

A normal is drawn at a

P\left( {x,y} \right)

of a curve. It meets the

x

-axis at

Q

. if

PQ

is of constant length

k

, then the differential equation describing such a curve is

View answer and explanation

Correct option: A

y\frac{{dy}}{{dx}} = \pm \sqrt {{k^2} - {y^2}}

Option A (

y\frac{{dy}}{{dx}} = \pm \sqrt {{k^2} - {y^2}}

) is correct because it represents the differential equation relating the length of the normal to the y-coordinate.

353 Medium

353 of 452

Application +4 | -1

y = A{e^x} + B{e^{2x}} + C{e^{3x}}

satisfies the differential equation

View answer and explanation

Correct option: A

y''' - 6y' + 11y' - 6y = 0

Option A (

y''' - 6y'' + 11y' - 6y = 0

) is correct because it is the differential equation that satisfies the given function

y = A{e^x} + B{e^{2x}} + C{e^{3x}}

354 Medium

354 of 452

Application +4 | -1

The solution of

\frac{{dy}}{{dx}} + y\tan x = \sec x

View answer and explanation

Correct option: A

y\sec x = \tan x + c

Option A (

y\sec x = \tan x + c

) is correct because it is the solution to the given first-order linear differential equation after finding the integrating factor and integrating.

355 Medium

355 of 452

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 obtained by separating variables and integrating the given differential equation.

356 Medium

356 of 452

Understanding +4 | -1

Which of the following equation is non-linear

View answer and explanation

Correct option: B

y \frac{d y}{d x}+4 x=0

Option B (

y \frac{{d y}}{{d x}}+4 x=0

) is non-linear because it involves the product of the dependent variable

y

and its derivative

\frac{{dy}}{{dx}}

357 Medium

357 of 452

Application +4 | -1

Consider the two statements :

Statement I

y=\sin k t

satisfy the differential equation

y^{\prime \prime}+9 y=0

.

Statement II

y=e^{k t}

satisfy the differential equation

y^{\prime \prime}+y^{\prime}-6 y=0

.

The value of

k

for which both the statements are correct is

View answer and explanation

Correct option: A

-3

Option A, -3, is the correct value of k for which both statements are correct, as it satisfies both differential equations.

358 Medium

358 of 452

Application +4 | -1

The equation of the curve in which subnormal varies as the square of the ordinate is (

\lambda

is constant of proportionality)

View answer and explanation

Correct option: B

y = C{e^{\lambda \mathrm{ }x}}

Option B (

y = C{e^{\lambda x}}

) is correct because it represents the solution of the differential equation derived from the given condition about the subnormal.

359 Medium

359 of 452

Application +4 | -1

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

where

y(0)=0

, then

y

is equal to:

View answer and explanation

Correct option: C

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

Option C (

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

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

360 Medium

360 of 452

Application +4 | -1

The differential equation of the curve for which the initial ordinate of any tangent is equal to the corresponding subnormal, is

View answer and explanation

Correct option: B

Homogeneous

Option B (Homogeneous) is correct because the resulting differential equation can be expressed in the form dy/dx = f(y/x), which is the definition of a homogeneous differential equation.

Revision notes and key takeaways

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