The solution $y^{n-1}\left(a y^{\prime}+y\right)=x$ is

$(n-1) y^{n-1}=C e^{-\frac{n x}{a}}+n x+a$

$n y^{n-1}=C e^{-\frac{n x}{a}}+n x$

$n y^{n-1}=C e^{-\frac{n x}{a}}+n x+a$

The solution of differential equation $\left( {1 + x} \right)y{\rm{ }}dx + \left( {1 - y} \right)x{\rm{ }}dy = 0$ is

${\log _e}\left( {\frac{x}{y}} \right) + x + y = c$

${\log _e}\left( {\frac{x}{y}} \right) - x + y = c$

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 · 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 931-960 of 975 questions

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931 Easy

931 of 975

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.

932 Medium

932 of 975

Application +4 | -1

A differential equation representing the family of parabolas with axis parallel to

y

-axis and whose length of latusrectum is the distance from the point origin to the line

4 x-3 y+5=0

is given by

View answer and explanation

Correct option: B

\frac{\mathrm{d}^2 y}{\mathrm{dx}^2}=2

Option B,

\frac{\mathrm{d}^2 y}{\mathrm{dx}^2}=2

, is the correct differential equation representing the family of parabolas with the given conditions.

933 Easy

933 of 975

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.

934 Easy

934 of 975

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.

935 Easy

935 of 975

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.

936 Easy

936 of 975

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.

937 Easy

937 of 975

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.

938 Easy

938 of 975

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.

939 Easy

939 of 975

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.

940 Medium

940 of 975

Application +4 | -1

The general solution of the differential equation

\sec ^2 x \tan y d x+\sec ^2 y \tan x d y=0

x, y \in\left(0, \frac{\pi}{4}\right)

View answer and explanation

Correct option: B

\cot x \cot y=c

Option C (

\cot x \cot y = c

) is incorrect. The correct answer is

\tan x \tan y = c

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

941 Medium

941 of 975

Application +4 | -1

The solution

y^{n-1}\left(a y^{\prime}+y\right)=x

View answer and explanation

Correct option: C

n y^n=C e^{-\frac{n x}{a}}+n x-a

Option C (

n y^n=C e^{-\frac{{n x}}{{a}}}+n x-a

) is correct because it is the solution obtained by making the substitution

u = y^n

and solving the resulting linear differential equation.

942 Easy

942 of 975

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.

943 Medium

943 of 975

Application +4 | -1

Solution of the differential equation

y\left(a x y+e^x\right) d x-e^x d y=0

where

y(0)=1

View answer and explanation

Correct option: A

y\left(2-a x^2\right)=2 e^x

Option A (

y(2-ax^2)=2e^x

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

944 Medium

944 of 975

Application +4 | -1

The solution of

\frac{d y}{d x}+\frac{y}{3}=1

View answer and explanation

Correct option: B

y=3+c e^{-x / 3}

Option B (

y=3+c e^{-x / 3}

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

945 Medium

945 of 975

Application +4 | -1

The solution of differential equation

\left( {1 + x} \right)y\mathrm{ }dx + \left( {1 - y} \right)x\mathrm{ }dy = 0

View answer and explanation

Correct option: A

{\log _e}\left( {xy} \right) + x - y = c

Option A (

{\log _e}\left( {xy} \right) + x - y = c

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

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946 Medium

946 of 975

Application +4 | -1

A function

y = f\left( x \right)

has a second order derivative

f'' = 6\left( {x - 1} \right)

. If its graph passes through the point (2,1) and at point the tangent to the graph is

y = 3x - 5

then the function is

View answer and explanation

Correct option: B

{\left( {x - 1} \right)^3}

Option B (

{(x - 1)^3}

) is correct because it satisfies the given conditions for the second derivative, the point (2,1), and the tangent line

y = 3x - 5

947 Easy

947 of 975

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.

948 Easy

948 of 975

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

949 Medium

949 of 975

Analyzing +4 | -1

The solution to the differential equation

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

where

f\left( x \right)

is a given function is

View answer and explanation

Correct option: A

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

Option A is correct. The solution to the differential equation is found by recognizing it as a Bernoulli equation and solving accordingly.

950 Medium

950 of 975

Application +4 | -1

The general solution of the differential equation

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

View answer and explanation

Correct option: C

x^2+2 x y=c

Option C (

x^2+2 x y=c

) is the correct general solution of the given differential equation, obtained by recognizing it as a homogeneous equation and applying the substitution

y=vx

951 Easy

951 of 975

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.

952 Easy

952 of 975

Application +4 | -1

The integrating factor of differential equation

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

View answer and explanation

Correct option: B

(1+x) e^{-x}

Option B (

(1+x)e^{-x}

) is correct because it is the integrating factor of the given differential equation after rewriting it in standard form.

953 Medium

953 of 975

Application +4 | -1

The solution of the differential equation

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

View answer and explanation

Correct option: A

\frac{x}{{\sin y}} + \log x = c

Option A (

\frac{x}{{\sin y}} + \log x = c

) is correct because it is the solution of the given differential equation after applying appropriate substitutions and integrating factors.

954 Medium

954 of 975

Application +4 | -1

y=0

when

x=1

, then a particular solution of the equation

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

View answer and explanation

Correct option: C

y\left(1+x^2\right)=\frac{\pi}{4}-\operatorname{Cot}^{-1} x

Option D (

y(1+x^2)=\operatorname{{Tan}}^{-1} x+\frac{{\pi}}{{4}}

) is incorrect. The correct answer is C (

y(1+x^2)=\frac{{\pi}}{{4}}-\operatorname{{Cot}}^{-1} x

). The error likely stems from incorrect handling of the initial condition or trigonometric identities.

955 Easy

955 of 975

Application +4 | -1

Solution of

x \frac{d y}{d x}+y-x+x y \cot x=0 \,\,\,(x \neq 0)

View answer and explanation

Correct option: A

x y=\frac{c}{\sin x}+1-x \cot x

Option A (

x y=\frac{{c}}{{\sin x}}+1-x \cot x

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

956 Medium

956 of 975

Application +4 | -1

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

Option C (

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

) is correct because it is the integrating factor obtained after rewriting the differential equation in standard form.

957 Medium

957 of 975

Application +4 | -1

The substitution

y=z^\alpha

transforms the differential equation

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

into a homogeneous differential equation for

View answer and explanation

Correct option: A

\alpha=-1

Option A is correct because the substitution transforms the given differential equation into a homogeneous equation when alpha is -1.

958 Medium

958 of 975

Application +4 | -1

The solution of the differential equation

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

is (where

{C}

is the constant of integration)

View answer and explanation

Correct option: B

\sin ^2 x \cdot y^2=x^2+C

Option B (

\sin ^2 x \cdot y^2=x^2+C

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

959 Easy

959 of 975

Application +4 | -1

Solution of

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

View answer and explanation

Correct option: A

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

Option A (

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

) is correct because the given differential equation is separable, and integrating both sides leads to this solution.

960 Easy

960 of 975

Understanding +4 | -1

The solution of

\frac{d y}{d x}+p(x) y=0

View answer and explanation

Correct option: C

y=c e^{-\int p d x}

Option C (

y=c e^{-\int p d x}

) is the correct solution to the given first-order linear homogeneous differential equation, derived by integrating factor method.

Revision notes and key takeaways

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