Let $y=y(x)$ be a solution curve of the differential equation. $\left(1-x^2 y^2\right) d x=y d x+x d y$ If the line $x=1$ intersects the curve $y=y(x)$ at $y=2$ and the line $x=2$ intersects the curve $y=y(x)$ at $y=\alpha$, then a value of $\alpha$ is

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

$\frac{3 e^2}{2\left(3 e^2+1\right)}$

A continuously differential function $\phi \left( x \right)$ in $\left( {0,\pi } \right)$ satisfying $y' = 1 + {y^2},y\left( 0 \right) = 0 = y\left( \pi \right)$, is

$\left( {x - \pi } \right)\left( {1 - {e^x}} \right)$

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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Medium

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361 Medium

361 of 452

Application +4 | -1

y' = \frac{{x - y}}{{x + y}}

, then its solution is

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it is the solution obtained by solving the homogeneous differential equation using appropriate substitution and integration.

362 Medium

362 of 452

Application +4 | -1

The equation of family of curves for which the length of the normal is equal to the radius vector, is

View answer and explanation

Correct option: A

{y^2} \mp {x^2} = {k^2}

Option A (

{y^2} \mp {x^2} = {k^2}

) is correct because it is the solution to the differential equation derived from the given condition about the length of the normal and the radius vector.

363 Medium

363 of 452

Application +4 | -1

\frac{d y}{d x}=y+\int\limits_{0}^1 y d x

and

y(0)=1

then

y(2)=

View answer and explanation

Correct option: D

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

Option C (

\sqrt{6}

) is correct because the maximum value of the scalar triple product occurs when

\bar{w}

is in the direction of

\bar{u} \times \bar{v}

, and its magnitude is

|\bar{u} \times \bar{v}|

364 Medium

364 of 452

Application +4 | -1

The solution of the equation

\frac{d y}{d x}+x(x+y)=x^{3}(x+y)^{3}-1

View answer and explanation

Correct option: C

\frac{1}{(x+y)^{2}}=x^{2}+1+c e^{x^{2}}

Option C (

\frac{{1}}{{(x+y)^{2}}}=x^{2}+1+c e^{x^{2}}

) is correct because it's the solution obtained by transforming the given differential equation into a Bernoulli equation and solving it.

365 Medium

365 of 452

Understanding +4 | -1

The order and the degree of differential equation

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

are .... and .... respectively.

View answer and explanation

Correct option: C

2,2

Option C (2,2) is correct because the highest order derivative is 2 and its power is 2 after squaring both sides.

366 Medium

366 of 452

Application +4 | -1

Let

y=y(x)

be a solution curve of the differential equation.

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

If the line

x=1

intersects the curve

y=y(x)

y=2

and the line

x=2

intersects the curve

y=y(x)

y=\alpha

, then a value of

\alpha

View answer and explanation

Correct option: B

\frac{1+3 e^2}{2\left(3 e^2-1\right)}

Option B (

\frac{{1+3 e^2}}{{2(3 e^2-1)}}}

) is correct because it represents the value of

\alpha

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

367 Medium

367 of 452

Application +4 | -1

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

, then its solution is

View answer and explanation

Correct option: A

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

Option A (

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

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

y=vx

and integrating.

368 Medium

368 of 452

Application +4 | -1

The solution of the differential equation

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

is (where

{C}

is the constant of integration)

View answer and explanation

Correct option: C

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

Option C (

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

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

369 Medium

369 of 452

Analyzing +4 | -1

Let

f:[0,1] \to R

be such that

f(xy) = f(x).f(y)

, for all

x,y \in [0,1]

and

f(0) \ne 0

. If

y = y(x)

satisfies the differential equation,

\frac{{dy}}{{dx}} = f(x)\mathrm{ }

with

y(0) = 1

, then

y\left( {\frac{1}{4}} \right) + y\left( {\frac{3}{4}} \right)

is equal to

View answer and explanation

Correct option: B

Option B (3) is correct because by using the given functional equation and the differential equation, we find that y = x + 1, and thus y(1/4) + y(3/4) = 3.

370 Medium

370 of 452

Application +4 | -1

The general solution of the equation

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

View answer and explanation

Correct option: B

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

Option B (

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

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

y=vx

371 Medium

371 of 452

Application +4 | -1

The solution of the differential equation

9y\frac{{dy}}{{dx}} + 4x = 0

View answer and explanation

Correct option: B

\frac{{{y^2}}}{4} + \frac{{{x^2}}}{9} = c

Option B (

\frac{{{y^2}}}{4} + \frac{{{x^2}}}{9} = c

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

372 Medium

372 of 452

Application +4 | -1

The solution of the differential equation

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

satisfying

y(1)=1

, is

View answer and explanation

Correct option: A

A hyperbola

The correct answer is A (A hyperbola) because the solution to the differential equation is a hyperbola.

373 Medium

373 of 452

Application +4 | -1

\phi(x)

is a differentiable function then the solution of

\mathrm{d} y+\left(y \phi^{\prime}(x)-\phi(x) \phi^{\prime}(x)\right) \mathrm{d} x=0

View answer and explanation

Correct option: A

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

Option A (

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

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

374 Medium

374 of 452

Application +4 | -1

The solution of differential equation

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

(where

{C}

is an arbitrary constant)

View answer and explanation

Correct option: B

3 x^4 y^2+y^3-x^3=C

Option B (

3 x^4 y^2+y^3-x^3=C

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

375 Medium

375 of 452

Application +4 | -1

The solution of the differential equation

x^2 \frac{d y}{d x} \cdot \cos \frac{1}{x}-y \sin \frac{1}{x}=-1,

where

y \rightarrow-1

x \rightarrow \infty

View answer and explanation

Correct option: A

y=\sin \frac{1}{x}-\cos \frac{1}{x}

Option A (

y=\sin \frac{1}{x}-\cos \frac{1}{x}

) is the correct solution obtained by solving the given differential equation using an integrating factor and applying the given condition.

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376 Medium

376 of 452

Application +4 | -1

Solution of the differential equation

(x-y)^2 \frac{\mathrm{d} y}{\mathrm{~d} x}=

a^2

View answer and explanation

Correct option: A

y=\frac{a}{2} \log \left|\frac{x-y-a}{x-y+a}\right|+c

Option A,

y=\frac{a}{2} \log \left|\frac{x-y-a}{x-y+a}\right|+c

, is the correct solution obtained by substituting

x-y=u

and integrating the resulting equation.

377 Medium

377 of 452

Application +4 | -1

The equation to the orthogonal trajectories of the system of parabolas

y=a x^2

View answer and explanation

Correct option: A

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

Option A is correct because the orthogonal trajectories are found by replacing dy/dx with -dx/dy and solving the resulting differential equation.

378 Medium

378 of 452

Application +4 | -1

The solution of the differential equation

x^3 \frac{d y}{d x}+4 x^2 \tan y=e^x \sec y

satisfying

y(1)=0

, is

View answer and explanation

Correct option: B

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

Option D (

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

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

379 Medium

379 of 452

Application +4 | -1

The differential equation obtained on eliminating

A

and

B

from the equation

y = A\cos \omega t + B\sin \omega t

View answer and explanation

Correct option: A

{y_2} = - {\omega ^2}y

Option A (

y_2 = - {\omega ^2}y

) is correct because it is the differential equation obtained by eliminating A and B from the given equation.

380 Medium

380 of 452

Application +4 | -1

y=f(x)

satisfies the differential equation

\sin x \frac{d y}{d x}+2 y \cos x=8

with

f(\pi / 2)=8

, then the minimum value of

f(x)

is:

View answer and explanation

Correct option: A

Option A (4) is correct because the minimum value of the function

f(x)

is found by solving the differential equation and analyzing the resulting function.

381 Medium

381 of 452

Application +4 | -1

The integrating factor of the differential equation

\frac{{dy}}{{dx}} + \frac{1}{x}.y = 3x

View answer and explanation

Correct option: A

x

Option A (

x

) is correct because it is the integrating factor calculated using the formula

e^{\int P dx}

for the given differential equation.

382 Medium

382 of 452

Application +4 | -1

A particular solution of the initial value differential equation

\log \frac{d y}{d x}=7 x+8 y, y(0)=0

View answer and explanation

Correct option: D

7 e^{-8 y}+8 e^{7 x}=15

The correct answer is C (

8 e^{-8 y}+7 e^{7 x}=15

) because it satisfies the given differential equation and the initial condition y(0)=0.

383 Medium

383 of 452

Application +4 | -1

Let

y=y(x)

be the solution of the differential equation

\sec x d y+\{2(1-x) \tan x+x(2-x)\} d x=0

such that

y(0)=2

. Then

y(2)

is equal to :

View answer and explanation

Correct option: D

The correct answer is D (2) because solving the differential equation and applying the initial condition leads to y(2) = 2.

384 Medium

384 of 452

Application +4 | -1

The differential equation of all circles which passes through the origin and whose centre lies on

y

-axis, is

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because it represents the differential equation of all circles passing through the origin with centers on the y-axis.

385 Medium

385 of 452

Application +4 | -1

The curve passing through the point

(0, \pi / 4)

satisfying the differential equation

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

View answer and explanation

Correct option: C

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

Option D (

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

) is the correct solution to the differential equation, obtained by transforming it into a linear form and applying the initial condition.

386 Medium

386 of 452

Analyzing +4 | -1

A continuously differential function

\phi \left( x \right)

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

satisfying

y' = 1 + {y^2},y\left( 0 \right) = 0 = y\left( \pi \right)

, is

View answer and explanation

Correct option: D

Not possible

Option D (Not possible) is correct because the solution to the differential equation is not continuous in the given interval.

387 Medium

387 of 452

Application +4 | -1

Solution of the differential equation

\cos xdy = y\left( {\sin x - y} \right)dx,0 < x < \frac{\pi }{2}

, is

View answer and explanation

Correct option: A

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

Option A (

\sec x = (\tan x + c)y

) is correct because it represents the solution obtained after solving the given differential equation using an integrating factor.

388 Medium

388 of 452

Application +4 | -1

A particular solution of

\log \left( {\frac{{dy}}{{dx}}} \right) = 3x + 4y,y\left( 0 \right) = 0

View answer and explanation

Correct option: C

3{e^{3x}} + 4{e^{ - 4y}} = 7

Option D (

4{e^{3x}} + 3{e^{ - 4y}} = 7

) is correct because it is the particular solution of the given differential equation that satisfies the initial condition y(0) = 0.

389 Medium

389 of 452

Application +4 | -1

The solution of

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

is given by

View answer and explanation

Correct option: D

x=C e^{2 y}+\frac{1}{2} y^2+\frac{1}{2} y+\frac{1}{4}

Option D (

x=C e^{2 y}+\frac{1}{2} y^2+\frac{1}{2} y+\frac{1}{4}

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

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

390 Medium

390 of 452

Application +4 | -1

Solution of the equation

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

View answer and explanation

Correct option: B

x y+y \log y-y=c

Option B (

x y+y \log y-y=c

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

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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