Which of the following is true for y(x) that satisfies the differential equation $\frac{{dy}}{{dx}} = {\rm{ }}xy{\rm{ }}-{\rm{ }}1{\rm{ }} + {\rm{ }}x{\rm{ }}-{\rm{ }}y;{\rm{ }}y{\rm{ }}\left( 0 \right){\rm{ }} = 0$

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

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

\[y{\rm{ }}\left( 1 \right){\rm{ }} = {e^{\frac{1}{2}}} - 1\]

The solution of $\frac{{dy}}{{dx}} + y = {e^{ - x}},y\left( 0 \right) = 0$ is

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

The solution of the equation $\frac{{{d^2}y}}{{d{x^2}}} = {e^{ - 2x}}$ is

$\frac{1}{4}{e^{ - 2x}} + c{x^2} + d$

$\frac{1}{4}{e^{ - 2x}} + c + d$

Solution of the differential equation $x{\rm{ }}dy - y{\rm{ }}dx = 0$ represents

A parabola whose vertex is at the origin

A circle whose centre is at the origin

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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 31-60 of 494 questions

31 Easy

31 of 494

Application +4 | -1

The differential equation of family of the curves

x^2+y^2=2 a x

is ..... . (

a

is an arbitrary constant)

View answer and explanation

Correct option: B

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

Option B (

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

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

32 Easy

32 of 494

Application +4 | -1

Solution of differential equation

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

View answer and explanation

Correct option: A

x=3 y^2+c y

Option A,

x=3 y^2+c y

, is the correct solution obtained by rearranging the equation and solving the differential equation.

33 Easy

33 of 494

Application +4 | -1

\phi \left( x \right) = \phi '\left( x \right)

and

\phi \left( 1 \right) = 2

, then

\phi \left( 3 \right)

equals

View answer and explanation

Correct option: B

2{e^2}

Option B (

2{e^2}

) is correct because it is the value of

\phi(3)

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

34 Easy

34 of 494

Application +4 | -1

A curve that passes through

(2,4)

and having subnormal of constant length of 8 units can be

View answer and explanation

Correct option: A

y^2=16 x-8

Option A (

y^2=16 x-8

) is the correct answer because it satisfies the given condition of having a constant subnormal length of 8 units and passes through the point (2,4).

35 Easy

35 of 494

Application +4 | -1

If a curve passes through the point

\left(2, \frac{7}{2}\right)

and has slope

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

at any point

(x, y)

on it, then the ordinate of the point on the curve whose abscissa is -2 is :

View answer and explanation

Correct option: A

-\frac{3}{2}

Option A (

-\frac{3}{2}

) is correct because it is the ordinate of the point on the curve whose abscissa is -2, obtained after solving the given differential equation and applying the initial condition.

36 Easy

36 of 494

Application +4 | -1

The solution of the differential equation

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

, is

View answer and explanation

Correct option: D

y\cos x = c + \frac{1}{2}\cos 2x

Option B (

y\cos x = c + \frac{1}{2}\sin 2x

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

37 Easy

37 of 494

Application +4 | -1

The equation of the curve through the point (3,2) and whose slope

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

, is

View answer and explanation

Correct option: D

\frac{{{y^2}}}{2} + y = \frac{{{x^3}}}{3} - 5

Option D (

\frac{{{y^2}}}{2} + y = \frac{{{x^3}}}{3} - 5

) is correct because it is the solution to the differential equation that passes through the point (3,2).

38 Easy

38 of 494

Application +4 | -1

\int \frac{\left(x^4-x\right)^{1 / 4}}{x^5} d x

is equal to : (where

\mathrm{c}

is constant of integration)

View answer and explanation

Correct option: A

\frac{4}{15}\left(1-\frac{1}{x^3}\right)^{5 / 4}+c

Option A (

\frac{{4}}{{15}}\left(1-\frac{{1}}{{x^3}}\right)^{{5 / 4}}+c

) is correct because it is the result of the integration after a suitable substitution.

39 Easy

39 of 494

Application +4 | -1

Which of the following is true for y(x) that satisfies the differential equation

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

View answer and explanation

Correct option: A

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

Option A (

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

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

40 Easy

40 of 494

Understanding +4 | -1

The order of the differential equation of the family of rectangular hyperbolas having fixed centre, is

View answer and explanation

Correct option: A

Option 2 is correct because the family of rectangular hyperbolas with a fixed center requires two parameters to define it.

41 Easy

41 of 494

Application +4 | -1

Solution of the differential equation

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

, is

View answer and explanation

Correct option: A

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

Option A (

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

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

42 Easy

42 of 494

Application +4 | -1

Let

y=y(x)

be the solution of the differential equation

\cos x \frac{d y}{d x}+y \sin x=1, x \in\left(0, \frac{\pi}{2}\right)

. If

{y}\left(\frac{\pi}{4}\right)=0

, then

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

View answer and explanation

Correct option: C

\frac{1-\sqrt{3}}{2}

Option C (

\frac{{1-\sqrt{{3}}}}{{2}}

) is correct because it is the value of y(

\frac{{\pi}}{{6}}

) obtained after solving the given differential equation and applying the initial condition.

43 Easy

43 of 494

Application +4 | -1

The solution of

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

View answer and explanation

Correct option: D

y = x{e^{ - x}}

Option D (

y = x{e^{ - x}}

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

44 Easy

44 of 494

Application +4 | -1

Suppose

y

is a function of

x

that satisfies

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

and

y=0

x=\frac{2}{\pi}

, then

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

is equal to:

View answer and explanation

Correct option: B

\frac{1}{2}

Option B (

\frac{1}{2}

) is correct because it's found by solving the differential equation and using the given initial condition to find the constant of integration.

45 Easy

45 of 494

Analyzing +4 | -1

The differential equation

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

determines a family of circles with

View answer and explanation

Correct option: C

Fixed radius 1 and variable centre along the

x

-axis

Option C (Fixed radius 1 and variable centre along the

x

-axis) is correct because the solution to the differential equation represents a family of circles with radius 1 and centers on the x-axis.

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46 Easy

46 of 494

Application +4 | -1

The solution of the equation

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

View answer and explanation

Correct option: B

\frac{{{e^{ - 2x}}}}{4} + cx + d

Option B (

\frac{{{e^{ - 2x}}}}{4} + cx + d

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

47 Easy

47 of 494

Application +4 | -1

The order and the degree of

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

are ..... and ......

View answer and explanation

Correct option: A

1,1

Option A is correct because the given equation can be rewritten as dy/dx = log(2x), which is a first-order, first-degree differential equation.

48 Easy

48 of 494

Application +4 | -1

The solution of initial value problem

x y^{\prime}+y

=0, y(2)=-2

is given by

View answer and explanation

Correct option: A

x y+4=0

Option A (

x y+4=0

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

x y^{\prime}+y=0

with the initial condition

y(2)=-2

49 Easy

49 of 494

Application +4 | -1

The solution of differential equation

2 x y \sec ^2\left(y^2\right) \frac{d y}{d x}+y^4=\ln \left(x^2 e^{y^4}\right)

View answer and explanation

Correct option: C

\tan y^2=(\ell \ln x)^2+C

Option C (

\tan y^2=(\ell n x)^2+C

) is correct because it is the solution obtained by substituting

y^2 = t

and integrating the resulting differential equation.

50 Easy

50 of 494

Application +4 | -1

The differential equation

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

has the solution

View answer and explanation

Correct option: A

x=y(\log x+C)

The correct answer is A (

x=y(\log x+C)

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

y=vx

51 Easy

51 of 494

Application +4 | -1

The solution of the equation

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

is given by

View answer and explanation

Correct option: B

y=C x-1

The correct answer is B (

y=C x-1

) because it is the general solution to the given separable differential equation.

52 Easy

52 of 494

Application +4 | -1

y=y(x)

is a function of

x

and

y(0)=0

, then a solution of the equation

\frac{d y}{d x}-y \tan x=\sec x

View answer and explanation

Correct option: A

y=x \sec x

Option B (

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

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

53 Easy

53 of 494

Application +4 | -1

The solution of the differential equation

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

, given that

y=1, \frac{d y}{d x}=0

when

x=1

, is

View answer and explanation

Correct option: B

y=x \log x-x+2

Option B (

y=x \log x-x+2

) is correct because it is the solution obtained by solving the given second-order differential equation with the given initial conditions.

54 Easy

54 of 494

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 is the solution obtained after integrating the given differential equation after separation of variables.

55 Easy

55 of 494

Application +4 | -1

Solution of the differential equation

x\mathrm{ }dy - y\mathrm{ }dx = 0

represents

View answer and explanation

Correct option: D

Straight lines passing through the origin

Option D (Straight lines passing through the origin) is correct because the solution to the differential equation is of the form y = Cx, which represents a family of straight lines passing through the origin.

56 Easy

56 of 494

Application +4 | -1

\frac{{dy}}{{dx}}\, = \,\frac{{{2^{x + y}} - {2^x}}}{{{2^y}}},\,y(0)\, = \,1

then

y(1)

is equal to

View answer and explanation

Correct option: B

{\log _2}(1 + e)

Option B (

{\log _2}(1 + e)

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

57 Easy

57 of 494

Understanding +4 | -1

The order and the degree of the differential equation

\frac{d^2 y}{d x^2}=\left[y+\left(\frac{d y}{d x}\right)^2\right]^{\frac{1}{4}}

are ...... and respectively .

View answer and explanation

Correct option: D

2 and 4

Option D (2 and 4) is correct because after raising both sides to the power of 4, the order is 2 and the degree is 4.

58 Easy

58 of 494

Understanding +4 | -1

The differential equation of all circles of radius

a

is of order

View answer and explanation

Correct option: A

Option A (2) is correct because the general equation of a circle with radius

a

has two arbitrary constants, so the order of the differential equation is 2.

59 Easy

59 of 494

Application +4 | -1

\frac{{dy}}{{dx}} + \frac{{{2^{x - y}}({2^y} - 1)}}{{{2^x} - 1}} = 0,x,y > 0,y(1) = 1

then

y(2)

is equal to

View answer and explanation

Correct option: D

2 - {\log _2}3

Option D (

2 - {\log _2}3

) is correct because by solving the differential equation and applying the initial condition, we can find the value of

y(2)

60 Easy

60 of 494

Understanding +4 | -1

The differential equation of all parabolas whose axis of symmetry is along

x

-axis is of order

View answer and explanation

Correct option: A

Option A (2) is correct because the general equation of the parabolas whose axis of symmetry is along the x-axis is

x=a y^2+b

, where

a, b

are arbitrary constants. So, the differential equation is of order 2.

Revision notes and key takeaways

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