Back to DIFFERENTIAL EQUATIONS

Your Progress

0 / 494 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • DIFFERENTIAL EQUATIONS

Differential equations practice · Easy

Try a question, then open “View answer” to check it. Sign in to submit answers and save progress.

Practice format and chapter quiz

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.

Take a chapter quiz →

Showing 1-30 of 494 questions

1 Easy

1 of 494

Application +4 | -1

Let

y=y(t)

be a solution of the differential equation

\frac{d y}{d t}+\alpha y=\gamma e^{-\beta t}

where,

\alpha>0, \beta>0

and

\gamma>0

. Then

\lim \limits_{{x \rightarrow \infty}} y(t)

View answer and explanation

Correct option: B

Is 0

Option B (Is 0) is correct because as t approaches infinity, both terms in the general solution of the differential equation approach 0 due to the negative exponential terms.

2 Easy

2 of 494

Understanding +4 | -1

The order and degree of the following differential equation

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

are respectively

View answer and explanation

Correct option: A

3,2

Option A (3,2) is correct because the order is 3 (third derivative) and after squaring, the degree of the third derivative is 2.

3 Easy

3 of 494

Application +4 | -1

Solution of the differential equation

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

, is

View answer and explanation

Correct option: D

e^{x / y}=-\cos y+C

Option D (

e^{x / y}=-\cos y+C

) is the correct solution obtained by recognizing the exact form and integrating the given differential equation.

4 Easy

4 of 494

Analyzing +4 | -1

The order of the differential equation whose general solution is given by

y = \left( {{c_1} + {c_2}} \right)\cos (x + {c_3}) - {c_4}{e^{x + {c_5}}}

where

{c_1},{c_2},{c_3},{c_4},{c_5}

are arbitrary constants, is

View answer and explanation

Correct option: B

Option B (3) is correct because the general solution can be simplified to contain only three independent arbitrary constants, thus the order of the differential equation is 3.

5 Easy

5 of 494

Application +4 | -1

The curve

y=f(x)(f(x) \geq 0, f(0)=0)

bounding a curvilinear trapezoid with the base

[0, x]

, whose area is proportional to the

(n+1)

th power of

f(x) f(1)=1

is given by

View answer and explanation

Correct option: A

x=y^n

Option A (

x=y^n

) is correct because it satisfies the given condition that the area under the curve is proportional to the (n+1)th power of f(x).

6 Easy

6 of 494

Application +4 | -1

The solution of differential equation

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

View answer and explanation

Correct option: D

\tan x \cdot \tan y=k

Option D (

\tan x \cdot \tan y=k

) is correct because separating variables and integrating leads to this solution.

7 Easy

7 of 494

Understanding +4 | -1

The degree of the differential equation

\left(\frac{d^3 y}{d x^3}\right)^{2 / 3}+4-3 \frac{d^2 y}{d x^2}+5 \frac{d y}{d x}=0

, is

View answer and explanation

Correct option: B

Option B is correct because after raising both sides to the power of 3 to eliminate the fractional exponent, the highest order derivative is d³y/dx³, and its power is 2, hence the degree is 2.

8 Easy

8 of 494

Application +4 | -1

Solution of the differential equation

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

, is

View answer and explanation

Correct option: D

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

The correct answer is D (

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

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

y=vx

9 Easy

9 of 494

Application +4 | -1

Solution of differential equation

\frac{d y}{d x}+\frac{y}{x}=x^2 y^6

is of form

\frac{2}{x^5 y^5}=\frac{A}{x^2}+C

. Then find value of

A

View answer and explanation

Correct option: A

Option A (5) is correct because by solving the differential equation and comparing with the given form, the value of A is found to be 5.

10 Easy

10 of 494

Application +4 | -1

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

Y

-axis and whose length of latus rectum is the distance of the point (2, -3) from the line is given by

3x + 4y = 5

, is given by

View answer and explanation

Correct option: D

11\frac{{{d^2}y}}{{d{x^2}}} = 10

Option D (

11\frac{{{d^2}y}}{{d{x^2}}} = 10

) is correct because it represents the differential equation derived from the given family of parabolas after eliminating the arbitrary constants.

11 Easy

11 of 494

Understanding +4 | -1

p

and

q

are order and degree of differential equation

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

, then:

View answer and explanation

Correct option: D

p+q=8

Option A (a periodic function with period

4a

) is correct because the function repeats its values after an interval of

4a

, satisfying the definition of a periodic function.

12 Easy

12 of 494

Application +4 | -1

The curve that passes through the point

(2,3)

and has the property that the segment of any tangent to it lying between the coordinate axes is bisected by the point of contact, is given by:

View answer and explanation

Correct option: A

y=\frac{6}{x}

Option A (

y=\frac{{6}}{{x}}

) is correct because it satisfies the given geometric property and passes through the point (2,3).

13 Easy

13 of 494

Application +4 | -1

x(t)

is a solution of

(t+1) d x=\left(2 x+(t+1)^4\right)

d t, x(0)=2

then

\lim \limits_{{t \rightarrow 1}} x(t)

View answer and explanation

Correct option: C

Option C (14) is correct because it is the limit of the solution to the differential equation as t approaches 1, given the initial condition.

14 Easy

14 of 494

Application +4 | -1

y(t)

is a solution of

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

and

y(0)=1

, then

y(1)

is equal to

View answer and explanation

Correct option: A

-\frac{1}{2}

Option A (

y=x \sec x

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

15 Easy

15 of 494

Application +4 | -1

Which of the following is a homogeneous differential equations?

View answer and explanation

Correct option: D

x \sin \left(\frac{y}{x}\right) d y=\left\{y \sin \left(\frac{y}{x}\right)-x\right\} d x

Option D (

x \sin \left(\frac{{y}}{{x}}\right) d y=\left\{y \sin \left(\frac{{y}}{{x}}\right)-x\right\} d x

) is correct because it can be written in the form dy/dx = f(y/x), which is the definition of a homogeneous differential equation.

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

16 Easy

16 of 494

Application +4 | -1

The integrating factor of differential equation

\frac{d y}{d x}+y \tan x-\sec x=0

View answer and explanation

Correct option: B

\sec x

Option B (sec x) is correct because it is the integrating factor obtained by solving the differential equation after identifying P(x) as tan x.

17 Easy

17 of 494

Application +4 | -1

The solution of the differential equation

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

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the solution obtained through separation of variables and integration does not match any of the given options.

18 Easy

18 of 494

Application +4 | -1

Let the slope of the tangent to a curve

y = f\left( x \right)

\left( {x,y} \right)

be given by

2\,\,\tan x\left( {\cos x - y} \right).

If the curve passes through the point

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

then the value of

\int\limits_0^{\pi /2} y dx

is equal to:

View answer and explanation

Correct option: B

2 - \frac{\pi }{{\sqrt 2 }}

Option B (2 - π/√2) is correct because it is the value of the definite integral of y dx from 0 to π/2, where y is the solution to the given differential equation.

19 Easy

19 of 494

Application +4 | -1

Let

y = y\left( x \right)

be the solution of the differential equation

x\left( {1 - {x^2}} \right)\frac{{dy}}{{dx}} + \left( {3{x^2}y - y - 4{x^3}} \right) = 0,\,x > 1,

with

y\left( 2 \right) = - 2.

Then

y\left( 3 \right)

is equal to

View answer and explanation

Correct option: A

–18

Option A (–18) is correct because it is the value of y(3) obtained by solving the given differential equation with the initial condition y(2) = -2.

20 Easy

20 of 494

Application +4 | -1

The general solution of the differential equation

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

, is

View answer and explanation

Correct option: B

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

Option B (

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

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

21 Easy

21 of 494

Application +4 | -1

Let

y = y\,(x)

be the solution of the differential equation

(x - {x^3})dy = (y + y{x^2} - 3{x^4})dx,\,x > 2

. If

y(3) = 3

, then

y\,(4)

is equal to

View answer and explanation

Correct option: B

Option B (12) is correct. The solution involves solving a differential equation with an initial condition and then evaluating the solution at x=4.

22 Easy

22 of 494

Application +4 | -1

y\frac{{dy}}{{dx}} = x\left[ {\frac{{{y^2}}}{{{x^2}}} + \frac{{\phi \left( {\frac{{{y^2}}}{{{x^2}}}} \right)}}{{\phi '\left( {\frac{{{y^2}}}{{{x^2}}}} \right)}}} \right],{\mkern 1mu} {\mkern 1mu} x > 0,{\mkern 1mu} {\mkern 1mu} \phi > 0

and

y(1)=-1

then

\phi \left( {\frac{{{y^2}}}{4}} \right)

is equal to

View answer and explanation

Correct option: B

4\,\,\phi \,\,(1)

Option B (

4\,\,\phi \,\,(1)

) is correct because it is the solution obtained after solving the given differential equation and applying the initial condition.

23 Easy

23 of 494

Application +4 | -1

The integrating factor of the differential equation

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

View answer and explanation

Correct option: C

1 / x

Option C (

1/x

) is correct because after rewriting the equation in standard form, the integrating factor is calculated as

e^{\int P dx}

24 Easy

24 of 494

Application +4 | -1

A solution of the differential equation

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

View answer and explanation

Correct option: C

y=2 x-4

Option C (

y=2x-4

) is correct because it satisfies the given differential equation when substituted.

25 Easy

25 of 494

Application +4 | -1

If a curve passes through the origin and the slope of the tangent to it at any point

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

\frac{{\left( {{x^2} - 4x + y + 8} \right)}}{{x - 2}}

, then this curve also passes through the point

View answer and explanation

Correct option: D

\left( {5,\,5} \right)

Option D (5,5) is correct because the curve passes through this point after solving the differential equation.

26 Easy

26 of 494

Application +4 | -1

The slope of a curve at any point is the reciprocal of twice the ordinate at the point and it passes through the point (4, 3). The equation of the curve is

View answer and explanation

Correct option: C

{y^2} = x + 5

Option C (

{y^2} = x + 5

) is correct because it satisfies the given differential equation and passes through the point (4, 3).

27 Easy

27 of 494

Application +4 | -1

The solution of the differential equation

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

View answer and explanation

Correct option: A

\log \left( {\frac{x}{y}} \right) = \frac{1}{x} + \frac{1}{y} + c

Option A (

\log \left( {\frac{x}{y}} \right) = \frac{1}{x} + \frac{1}{y} + c

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

28 Easy

28 of 494

Application +4 | -1

Let

y=y(x)

be the solution curve of the differential equation

\frac{d y}{d x}+\left(\frac{2 x^2+11 x+13}{x^3+6 x^2+11 x+6}\right) y=\frac{(x+3)}{x+1}, x>-1

, which passes through the point

(0,1)

. Then

y(1)

is equal to

View answer and explanation

Correct option: B

\frac{3}{2}

Option B (

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

) is correct because after solving the linear differential equation and applying the initial condition, y(1) evaluates to this value.

29 Easy

29 of 494

Application +4 | -1

Let

f

be a real-valued on

\mathbf{R}

such that

f(1)=1

. If the

y

-intercept of the tangent at any point

\{(x, y)

on the curve

y=f(x)

is equal to cube of the abscissa of

P

, then the value of

f(-3)

is equal to

View answer and explanation

Correct option: D

Option D (9) is correct because it is the value of f(-3) obtained by solving the given differential equation and applying the initial condition f(1)=1.

30 Easy

30 of 494

Analyzing +4 | -1

The order of the differential equation of all tangent lines to the parabola

y = {x^2}

View answer and explanation

Correct option: A

Option A (1) is correct because the differential equation of all tangent lines to the parabola

y = x^2

is of order 1.

Revision notes and key takeaways

Review DIFFERENTIAL EQUATIONS in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 494 single-correct-answer MCQs on DIFFERENTIAL EQUATIONS.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

Showing 1-30 of 494 questions

Continue Your Practice

📚 Switch Set (EASY)

Chapter Sets

ALL 975 MCQs EASY 494 MCQs HARD 29 MCQs MEDIUM 452 MCQs

0 /494

0✓ 0✗

0% Accuracy