The solution of the equation $\frac{{dy}}{{dx}} = \frac{{x + y}}{{x - y}}$ is

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

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JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Easy

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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 451-480 of 494 questions

451 Easy

451 of 494

Application +4 | -1

y=\frac{x}{\ln |c x|}

(where

c

is an arbitrary constant) is the general solution of the differential equation

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

, then the function

\phi\left(\frac{x}{y}\right)

View answer and explanation

Correct option: D

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

Option D,

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

, is the correct function

\phi(\frac{x}{y})

obtained by differentiating the given solution and substituting it into the differential equation.

452 Easy

452 of 494

Understanding +4 | -1

Which of the following equations is a linear equation of order 3 ?

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it is a linear differential equation of order 3, where the dependent variable and its derivatives appear only to the first power and are not multiplied together.

453 Easy

453 of 494

Application +4 | -1

The degree of homogeneous function

f(x, y)=\frac{x^3-y^3}{x^2+y^2}

is ...... .

View answer and explanation

Correct option: A

Option A (1) is correct because the function can be expressed as x times a function of y/x, indicating a homogeneous function of degree 1.

454 Easy

454 of 494

Application +4 | -1

The solution of the differential equation

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

is (where

{C}

is the constant integration)

View answer and explanation

Correct option: D

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

Option D (

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

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

455 Easy

455 of 494

Application +4 | -1

The integrating factor of the differential equation

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

, is

View answer and explanation

Correct option: C

\sqrt{1-x^2}

Option C (

\sqrt{{1-x^2}}

) is correct because it is the integrating factor obtained from

e^{\int P dx}

where P is the coefficient of y in the standard form of the linear differential equation.

456 Easy

456 of 494

Application +4 | -1

The solution of the differential equation

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

View answer and explanation

Correct option: A

\phi \left( {\frac{y}{x}} \right) = kx

Option A,

\phi \left( {\frac{y}{x}} \right) = kx

, is the correct solution because it satisfies the given differential equation after substitution and integration.

457 Easy

457 of 494

Application +4 | -1

The solution of the equation

(2+x) d y-y \, d x=0

represents a curve passing through a fixed point

P

, then the area of equilateral triangle with

P

as one vertex and

x+y=0

as its one side, is

View answer and explanation

Correct option: C

\frac{2}{\sqrt{3}}

Option C (

\frac{2}{\sqrt{3}}

) is correct because the area of the equilateral triangle is calculated using the distance from the point P to the line x+y=0 as the altitude.

458 Easy

458 of 494

Application +4 | -1

y=a e^{m x}+b e^{-m x}

satisfies which of the following differential equations?

View answer and explanation

Correct option: D

\frac{d^2 y}{d x^2}-m^2 y=0

Option D (

\frac{d^2 y}{d x^2}-m^2 y=0

) is correct because it is the differential equation that results from differentiating the given function twice.

459 Easy

459 of 494

Application +4 | -1

A solution of the differential equation

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

View answer and explanation

Correct option: C

y = 2x - 4

Option C (

y = 2x - 4

) is a solution to the given differential equation, which can be verified by substitution and differentiation.

460 Easy

460 of 494

Application +4 | -1

x\sin \left( {\frac{y}{x}} \right)dy = \left[ {y\sin \left( {\frac{y}{x}} \right) - y} \right]dx

and

y\left( 1 \right) = \frac{\pi }{2}

, then the value of

\cos \left( {\frac{y}{x}} \right)

is equal to

View answer and explanation

Correct option: C

\log x

Option C (

\log x

) is correct because it is the value of

\cos(\frac{y}{x})

obtained by solving the homogeneous differential equation with the given initial condition.

461 Easy

461 of 494

Application +4 | -1

Let

y = y(x)

be the solution of the differential equation,

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

such that

y(0) = 0

. If

\sqrt a \,y(1) = \frac{\pi }{{32}},

then the value of

'a'

View answer and explanation

Correct option: D

\frac{1}{{16}}

Option D (

\frac{1}{{16}}

) is correct because it is the value of 'a' obtained by solving the differential equation and using the given conditions.

462 Easy

462 of 494

Application +4 | -1

The solution of

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

View answer and explanation

Correct option: A

y^2(\log y)-e^x \sin ^2 x+c=0

Option A (

y^2(\log y)-e^x \sin ^2 x+c=0

) is correct because it is the solution obtained by separating variables and integrating by parts.

463 Easy

463 of 494

Understanding +4 | -1

The degree of the equation

{e^x} + \sin \left( {\frac{{dy}}{{dx}}} \right) = 3

View answer and explanation

Correct option: C

Degree is not defined

The correct answer is C (Degree is not defined) because the equation involves a trigonometric function of a derivative, making it non-polynomial in derivatives.

464 Easy

464 of 494

Understanding +4 | -1

The solution of the differential equation

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

View answer and explanation

Correct option: A

xy = c

Option A (

xy = c

) is correct because integrating the given differential equation directly leads to this solution.

465 Easy

465 of 494

Understanding +4 | -1

The degree of differential equation

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

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the degree of the differential equation is not defined since the equation is not a polynomial in all differential coefficients due to the logarithm.

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466 Easy

466 of 494

Application +4 | -1

A curve passing through origin, all the normals to which pass through

\left(x_0, y_0\right)

View answer and explanation

Correct option: B

x^2+y^2=2\left(x x_0+y y_0\right)

Option B (

x^2+y^2=2\left(x x_0+y y_0\right)

) is correct because it represents a curve passing through the origin where all normals pass through (x0, y0).

467 Easy

467 of 494

Application +4 | -1

The curve such that the ratio of the subnormal at any point to the sum of its abscissa and ordinate is equal to the ratio of the ordinate of this point to its abscissa is

View answer and explanation

Correct option: D

y=x \log |C x|

Option D (

y=x \log |C x|

) is correct because it satisfies the given differential equation derived from the ratio of subnormal to the sum of abscissa and ordinate.

468 Easy

468 of 494

Understanding +4 | -1

To reduce the differential equation

\frac{{dy}}{{dx}} + P\left( x \right).y = Q\left( x \right).{y^n}

to the linear form, the substitution is

View answer and explanation

Correct option: B

v = \frac{1}{{{y^{n - 1}}}}

Option B (

v = \frac{{1}}{{{y^{n - 1}}}}}}

) is correct because this substitution transforms the Bernoulli equation into a linear differential equation.

469 Easy

469 of 494

Application +4 | -1

y = y(x)

is the solution of the differential equation

2{x^2}\frac{{dy}}{{dx}} - 2xy + 3{y^2} = 0

such that

y(e) = \frac{e}{3},

then

y(1)

is equal to

View answer and explanation

Correct option: B

02-Mar

Option B (02-Mar) which is equivalent to 2/3 is correct because it is the value of y(1) obtained by solving the given differential equation with the specified initial condition.

470 Easy

470 of 494

Understanding +4 | -1

The degree of the differential equation

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

View answer and explanation

Correct option: D

none of these

Option D (none of these) is correct because the degree of the differential equation is not defined since the equation is not a polynomial in all the differential coefficients.

471 Easy

471 of 494

Application +4 | -1

The differential equation corresponding to the family of curves

y=e^x(a \cos x+b \sin x), a

and

b

being arbitrary constants is

View answer and explanation

Correct option: B

y_2-2 y_1+2 y=0

Option B (

y_2-2 y_1+2 y=0

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

472 Easy

472 of 494

Application +4 | -1

y(x)

is the solution of the differential equation

(x+2) \frac{d y}{d x}=x^2+4 x-9, x \neq-2

and

y(0)=0

, then

y(-4)

is equal to:0

View answer and explanation

Correct option: A

Option A (0) is correct because it is the value of y(-4) obtained after solving the given differential equation and applying the initial condition y(0) = 0.

473 Easy

473 of 494

Application +4 | -1

The equation of a curve passing through the origin and satisfying the differential equation

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

, is

View answer and explanation

Correct option: A

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

Option A (

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

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

474 Easy

474 of 494

Analyzing +4 | -1

The differential equation representing the family of curves

y^2=2 c(x+\sqrt{c})

, where

c

is a positive parameter, is of

View answer and explanation

Correct option: A

Order 1 and degree 3

Option A (Order 1 and degree 3) is correct because after eliminating the arbitrary constant, the resulting differential equation has order 1 and degree 3.

475 Easy

475 of 494

Application +4 | -1

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

, then solution of the differential equation is.

View answer and explanation

Correct option: A

\sin x+c_1 x+c_2

Option A (

\sin x+c_1 x+c_2

) is correct because it is the general solution obtained by integrating the given second-order differential equation twice.

476 Easy

476 of 494

Application +4 | -1

The solution of the differential equation

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

, is

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because it is the solution obtained by rewriting the differential equation, finding the integrating factor, and integrating.

477 Easy

477 of 494

Application +4 | -1

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

has the solution

View answer and explanation

Correct option: A

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

Option A (

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

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

478 Easy

478 of 494

Analyzing +4 | -1

The differential equation representing the family of curves

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

, where

c

is a positive parameter, is of

View answer and explanation

Correct option: C

Order 1, degree 5

Option C (Order 1, degree 5) is correct because after eliminating the parameter

c

, the resulting differential equation is of order 1 and degree 5.

479 Easy

479 of 494

Application +4 | -1

x d y=y(d x+y d y), y > 0

and

y(1)=1

, then

y(-3)

is equal to

View answer and explanation

Correct option: B

Option B (3) is the correct value of

y(-3)

because after solving the differential equation and applying the initial condition

y(1)=1

, we find the solution and substitute

x=-3

480 Easy

480 of 494

Application +4 | -1

The solution of the equation

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

View answer and explanation

Correct option: B

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

Option B (

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

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