CUET Mathematics - Calculus - Differential Equations | Free Questions

Q1:

2026: 31 May Shift 2

Calculus

>

Differential Equations

Easy

Common

Solution of differential equation $x\,dy - y\,dx = 0$ represents:

Family of straight lines passing through origin

Family of parabolas whose vertex is at origin

Family of circles whose centre is at origin

Family of straight lines passing through (1,1)

Correct Answer

Option 1

Correct Answer

Explanation

Q2:

2026: 31 May Shift 2

Calculus

>

Differential Equations

Medium

Common

If $m$ and $n$ are respectively order and degree of the differential equation

$\left(\dfrac{d^2y}{dx^2}\right)^5 + 6\cdot\dfrac{\left(\dfrac{d^2y}{dx^2}\right)^3}{\dfrac{d^3y}{dx^3}} + \dfrac{d^3y}{dx^3} = x^2+5$ then :

m = 3, n = 3

m = 2, n = 3

m = 3, n = 2

m = 3, n = 5

Correct Answer

Option 3

Correct Answer

Explanation

Q3:

2026: 31 May Shift 2

Calculus

>

Differential Equations

Medium

Common

General solution of the differential equation $\dfrac{dy}{dx} = e^{ax+by}$ is: (where C is an arbitrary constant)

$e^{ax} + e^{-by} = c$

$e^{ax+by} = c$

$be^{ax} + ae^{-by} = c$

$ae^{ax} - be^{-by} = c$

Correct Answer

Option 3

Correct Answer

Explanation

Q4:

2026: 31 May Shift 2

Calculus

>

Differential Equations

Medium

Core

The general solution of the differential equation $x^2\dfrac{dy}{dx} + y^2 = xy$ is: (take $\log_e x = \log x$ )

$\dfrac{x}{y} = \log x + C$ , C is an arbitrary constant

$\dfrac{x}{y} = 2\log x + C$ , C is an arbitrary constant

$y = \log x + C$ , C is an arbitrary constant

$x = \log y + C$ , C is an arbitrary constant

Correct Answer

Option 1

Correct Answer

Explanation

Q5:

2026: 31 May Shift 2

Calculus

>

Differential Equations

Medium

Core

Match the LIST-I with LIST-II

	LIST-I Differential equation		LIST-II Integrating factor
A.	$\dfrac{dy}{dx} + 3y = e^{-2x}$	I.	$\dfrac{1}{x}$
B.	$x\dfrac{dy}{dx} + y = 3x^2$	II.	$e^{-x}$
C.	$x\dfrac{dy}{dx} - y = 3x^2$	III.	$x$
D.	$\dfrac{dy}{dx} - y = x$	IV.	$e^{3x}$

Choose the correct answer from the options given below:

A-IV, B-III, C-I, D-II

A-III, B-IV, C-I, D-II

A-II, B-III, C-IV, D-I

A-I, B-II, C-III, D-IV

Correct Answer

Option 1

Correct Answer

Explanation

Q6:

2026: 31 May Shift 2

Calculus

>

Differential Equations

Medium

Applied

Match the LIST-I with LIST-II

[ $a$ is an arbitrary constant]

	LIST-I Differential Equation		LIST-II Family of curves
A.	$x\,dy - y\,dx = 0$	I.	$x^2 + y^2 = a^2$
B.	$2y\,dx - x\,dy = 0$	II.	$y = ax$
C.	$x\,dx + y\,dy = 0$	III.	$y^2 = 16x + a$
D.	$y\,dy - 8\,dx = 0$	IV.	$x^2 = ay$

Choose the correct answer from the options given below:

A-III, B-I, C-IV, D-II

A-II, B-IV, C-I, D-III

A-IV, B-II, C-III, D-I

A-I, B-III, C-II, D-IV

Correct Answer

Option 2

Correct Answer

Explanation

Q7:

2026: 30 May Shift 2

Calculus

>

Differential Equations

Easy

Common

Match List-I with List-II

List-I	List-II
Curve $y=f(x)$ (where C is an arbitrary constant)	The curve $y=f(x)$ satisfies the differential equation
(A) $y = \log x + C$	(I) $x\frac{dy}{dx} - y = 0$
(B) $y = e^x + C$	(II) $x\frac{dy}{dx} - 1 = 0$
(C) $y = cx$	(III) $x^2\frac{dy}{dx} + 1 = 0$
(D) $y = \frac{1}{x} + C$	(IV) $\frac{d^2y}{dx^2} - \frac{dy}{dx} = 0$

Choose the correct answer from the options given below:

(A) - (III), (B) - (II), (C) - (I), (D) - (IV)

(A) - (II), (B) - (IV), (C) - (I), (D) - (III)

(A) - (II), (B) - (IV), (C) - (III), (D) - (I)

(A) - (III), (B) - (I), (C) - (IV), (D) - (II)

Correct Answer

Option 2

Correct Answer

Explanation

Q8:

2026: 30 May Shift 2

Calculus

>

Differential Equations

Medium

Common

If $y = x\frac{dy}{dx} + \sqrt{a^2\left(\frac{dy}{dx}\right)^2 + b^2}$ , then the order and degree of the differential equation are

order = 2, degree = 2

order = 2, degree = 1

order = 1, degree = 2

order = 2, degree = not defined

Correct Answer

Option 3

Correct Answer

Explanation

Q9:

2026: 30 May Shift 2

Calculus

>

Differential Equations

Medium

Common

The general solution of the differential equation $x(x^2-1)\frac{dy}{dx} = 1$ is:

[Where C is an arbitrary constant]

$y = \frac{1}{2}\log_e\left|\frac{x^2-1}{x^2}\right| + C$

$y = \log|x^2(x^2-1)| + C$

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

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

Correct Answer

Option 1

Correct Answer

Explanation

Q10:

2026: 30 May Shift 2

Calculus

>

Differential Equations

Medium

Core

Match List-I with List-II

List-I	List-II
Differential equation	Integrating factor
(A) $\frac{dx}{dy}+\frac{2x}{y}=y^2$	(I) $e^x$
(B) $\frac{dy}{dx}+y=e^{-x}$	(II) $\frac{1}{\sqrt{y}}$
(C) $\frac{dy}{dx}-\frac{y}{2}=x$	(III) $y^2$
(D) $\frac{dx}{dy}-\frac{1}{2}\frac{x}{y}=\sin y$	(IV) $e^{-x/2}$

Choose the correct answer from the options given below:

(A) - (III), (B) - (I), (C) - (IV), (D) - (II)

(A) - (III), (B) - (I), (C) - (II), (D) - (IV)

(A) - (III), (B) - (II), (C) - (I), (D) - (IV)

(A) - (I), (B) - (IV), (C) - (III), (D) - (II)

Correct Answer

Option 1

Correct Answer

Explanation

Q11:

2026: 30 May Shift 2

Calculus

>

Differential Equations

Medium

Core

The particular solution of the differential equation $\sec^2x\tan y\,dx - \sec^2y\tan x\,dy = 0$ , $y=\frac{\pi}{4}$ when $x=\frac{\pi}{4}$ , is

$\tan x\tan y=1$

$\tan x\tan y=-1$

$\tan xy=-\tan x$

$\tan y=\tan x$

Correct Answer

Option 4

Correct Answer

Explanation

Q12:

2026: 30 May Shift 2

Calculus

>

Differential Equations

Easy

Applied

The differential equation representing family of circles with centre at origin and radius r units is-

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

$\frac{dy}{dx}=\frac{2r}{y}$

$xdy=-ydx$

$ydy=-2xdx$

Correct Answer

Option 1

Correct Answer

Explanation

Q13:

2026: 30 May Shift 1

Calculus

>

Differential Equations

Easy

Common

Let $y(x)$ be the solution curve of the differential equation

$\frac{dy}{dx} + 2y^2 = 0, y(1) = 1$ , then $y\left(\frac{3}{4}\right)$ is equal to:

$\frac{1}{3}$

$3$

$2$

$\frac{2}{3}$

Correct Answer

Option 3

Correct Answer

Explanation

Q14:

2026: 30 May Shift 1

Calculus

>

Differential Equations

Easy

Common

Match the LIST-I with LIST-II

	LIST-I Differential equation		LIST-II Sum of the order and the degree of the differential equation
A.	$\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right) = 1$	I.	$2$
B.	$4 + \left(\dfrac{dy}{dx}\right)^4 = 7\left(\dfrac{d^2y}{dx^2}\right)^3$	II.	$3$
C.	$x^3\left(\dfrac{d^2y}{dx^2}\right)^2 + x\left(\dfrac{dy}{dx}\right)^4 + 1 = 0$	III.	$5$
D.	$\dfrac{dy}{dx} = x^4 e^{-3y}$	IV.	$4$

Choose the correct answer from the options given below:

A-II, B-III, C-IV, D-I

A-III, B-II, C-I, D-IV

A-IV, B-I, C-II, D-III

A-II, B-III, C-I, D-IV

Correct Answer

Option 1

Correct Answer

Explanation

Q15:

2026: 30 May Shift 1

Calculus

>

Differential Equations

Medium

Core

Integrating factor of the differential equation $x\log x\frac{dy}{dx}+y=2\log x$ is equal to:

(Consider $\log x=\log_e x$ )

$x\log x$

$\log|x|$

$e^x$

$\frac{1}{x}$

Correct Answer

Option 2

Correct Answer

Explanation

Q16:

2026: 30 May Shift 1

Calculus

>

Differential Equations

Medium

Core

The general solution of the differential equation $\frac{dy}{dx}-\frac{y}{x}+\text{cosec}\left(\frac{y}{x}\right)=0$ is

(Consider $\log x=\log_e x$ )

$\sin\left(\frac{x}{y}\right)=\log x+C$ , where C is an arbitrary constant

$\sin\left(\frac{y}{x}\right)=\log x+C$ , where C is an arbitrary constant

$\cos\left(\frac{y}{x}\right)=\log x+C$ , where C is an arbitrary constant

$\cos\left(\frac{x}{y}\right)=\log x+C$ , where C is an arbitrary constant

Correct Answer

Option 3

Correct Answer

Explanation

Q17:

2026: 30 May Shift 1

Calculus

>

Differential Equations

Medium

Applied

The particular solution of the differential equation $\frac{dy}{dx}=x(2\log x+1)$ , given that $y=0$ when $x=2$ , is:

(consider $\log x=\log_e x$ )

$y=x\log x-3\log2$

$y=x^2\log x-\log2$

$y=x^2\log x-4\log2$

$y=x^2\log x+3$

Correct Answer

Option 3

Correct Answer

Explanation

Q18:

2026: 29 May Shift 2

Calculus

>

Differential Equations

Easy

Common

The order of the differential equation $\left(\frac{d^3y}{dx^3}\right)^4 + \frac{d^4y}{dx^4} + \log\left(\frac{dy}{dx}\right) = 0$ is

4

3

1

Not defined

Correct Answer

Option 1

Correct Answer

Explanation

Q19:

2026: 29 May Shift 2

Calculus

>

Differential Equations

Medium

Common

The solution of the differential equation $\frac{dy}{dx} + y = 1, y(1) = 0$ is

$1+x = \log_e|1-y|$

$1-x = \log_e|1+y|$

$1+x = \log_e|1+y|$

$1-x = \log_e|1-y|$

Correct Answer

Option 4

Correct Answer

Explanation

Q20:

2026: 29 May Shift 2

Calculus

>

Differential Equations

Easy

Common

The general solution of the differential equation $x\,dx = (x^2+1)\,dy$ is (where C is an arbitrary constant)

$y = \log_e|x^2+1| + C$

$y = \frac{1}{2}\log_e|x^2+1| + C$

$\frac{y^2}{2} = \frac{x^3}{3} + x + C$

$y = \frac{x^2}{2} + \log_e|x| + C$

Correct Answer

Option 2

Correct Answer

Explanation

Q21:

2026: 29 May Shift 2

Calculus

>

Differential Equations

Hard

Core

Match the LIST-I with LIST-II

	LIST-I Differential equations		LIST-II Integrating factor
A.	$(x+y)\dfrac{dy}{dx} = 1$	I.	$e^{-x}$
B.	$x\dfrac{dy}{dx} - y = x^2$	II.	$x$
C.	$\dfrac{dy}{dx} - y = \sin x$	III.	$\dfrac{1}{x}$
D.	$x\dfrac{dy}{dx} + y = \tan x$	IV.	$e^{-y}$

Choose the correct answer from the options given below:

A - I, B - III, C - IV, D - II

A - I, B - II, C - IV, D - III

A - IV, B - I, C - III, D - II

A - IV, B - III, C - I, D - II

Correct Answer

Option 4

Correct Answer

Explanation

Q22:

2026: 29 May Shift 2

Calculus

>

Differential Equations

Medium

Core

For the differential equation $\frac{d}{dx}\left(\frac{d^3y}{dx^3}\right)+\left(\frac{dy}{dx}\right)^3+y+x=7$ , which of the following statements are True?

A. Order of differential equation is 4

B. Number of arbitrary constants in the particular solution of differential equation are 4

C. Number of arbitrary constants in the general solution of the differential equation are 4

D. Degree of differential equation is 3

Choose the correct answer from the options given below:

A and C only

B and D only

B and C only

A and D only

Correct Answer

Option 1

Correct Answer

Explanation

Q23:

2026: 29 May Shift 2

Calculus

>

Differential Equations

Medium

Applied

In reference to the differential equation : $y''+y=(e)^{y'}$ , which of the following statements are correct ?

A. The order of the differential equation is 1.

B. The degree of the differential equation is 1.

C. The order of the differential equation is 2.

D. The degree of the differential equation is not defined.

Choose the correct answer from the options given below:

A and B only

C and D only

B and C only

A and D only

Correct Answer

Option 2

Correct Answer

Explanation

Q24:

2026: 29 May Shift 1

Calculus

>

Differential Equations

Medium

Common

Match List-I with List-II

List-I (Differential Equation)	List-II (Degree)
(A) $\left(\dfrac{dy}{dx}\right)^3+\left(\dfrac{d^2y}{dx^2}\right)^2=0$	(I) Not defined
(B) $\left(\dfrac{d^2y}{dx^2}\right)^3+\left(\dfrac{dy}{dx}\right)^2+1=0$	(II) 2
(C) $\sqrt{1+\dfrac{d^2y}{dx^2}}=\dfrac{dy}{dx}+x$	(III) 1
(D) $\dfrac{d^2y}{dx^2}+e^{\frac{dy}{dx}}=0$	(IV) 3

Choose the correct answer from the options given below:

(A) - (II), (B) - (IV), (C) - (III), (D) - (I)

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

(A) - (IV), (B) - (I), (C) - (II), (D) - (III)

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

Correct Answer

Option 1

Correct Answer

Explanation

Q25:

2026: 29 May Shift 1

Calculus

>

Differential Equations

Medium

Common

The general solution of the differential equation $(x+1)\dfrac{dy}{dx}=2e^{-y}-1$ is:

(where C is an arbitrary constant)

$e^y(x+1)=C$

$x(e^y-2)=C$

$(x+1)(e^y-2)=C$

$xe^y+2x=C$

Correct Answer

Option 3

Correct Answer

Explanation

Q26:

2026: 29 May Shift 1

Calculus

>

Differential Equations

Easy

Common

The general solution of the differential equation $\dfrac{dy}{dx}=2^{-y}$ is:

(where $c$ is an arbitrary constant)

$2^y=x^2\log_e2+c$

$2^y=x\log_e2+cx$

$2^y=x\log_e2+c$

$2^y=-x\log_e2+c$

Correct Answer

Option 3

Correct Answer

Explanation

Q27:

2026: 29 May Shift 1

Calculus

>

Differential Equations

Medium

Core

The substitution $y=vx$ transforms the differential equation $xdy-ydx=\sqrt{x^2+y^2}dx$ into

$\dfrac{dv}{\sqrt{1+v^2}}=\dfrac{dx}{x}$

$\dfrac{dv}{\sqrt{1+v}}=\dfrac{dx}{x^2}$

$\dfrac{dv}{v}=\dfrac{dx}{\sqrt{1+x^2}}$

$\dfrac{dv}{\sqrt{1+v^2}}=\dfrac{dx}{1+x^2}$

Correct Answer

Option 1

Correct Answer

Explanation

Q28:

2026: 29 May Shift 1

Calculus

>

Differential Equations

Medium

Core

Match List-I with List-II

List-I (Differential equation)	List-II (Integrating factor)
(A) $\dfrac{xdy}{dx}-y=x^2e^x$	(I) $x$
(B) $\dfrac{dy}{dx}-2y=e^x$	(II) $x^4$
(C) $\dfrac{xdy}{dx}+y=1$	(III) $\dfrac{1}{x}$
(D) $\dfrac{xdy}{dx}+4y=x^4$	(IV) $e^{-2x}$

Choose the correct answer from the options given below:

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)

(A) - (I), (B) - (II), (C) - (III), (D) - (IV)

(A) - (III), (B) - (IV), (C) - (II), (D) - (I)

(A) - (I), (B) - (III), (C) - (II), (D) - (IV)

Correct Answer

Option 1

Correct Answer

Explanation

Q29:

2026: 29 May Shift 1

Calculus

>

Differential Equations

Medium

Applied

Which of the following are correct?

(A) The order of the differential equation $\left(\dfrac{dy}{dx}\right)^2+3y^2-5y=0$ is 2.

(B) The degree of the differential equation $x\dfrac{d^2y}{dx^2}=\left(3+2\left(\dfrac{dy}{dx}\right)^2\right)^3$ is 3.

(C) The differential equation representing the curve $y=cx$ , $c$ being an arbitrary constant is $ydx-xdy=0$ .

(D) The general solution of an ordinary differential equation of $n^{th}$ order contains $n$ independent arbitrary constants.

Choose the correct answer from the options given below:

(A), (B) and (C) only

(A), (C) and (D) only

(B), (C) and (D) only

(C) and (D) only

Correct Answer

Option 4

Correct Answer

Explanation

Q30:

2026: 26 May Shift 2

Calculus

>

Differential Equations

Medium

Common

Match the LIST-I with LIST-II

	LIST-I Differential equation		LIST-II Order and Degree
A.	$\left(\dfrac{dy}{dx}\right)^2 + 3y\left(\dfrac{d^2y}{dx^2}\right) = 0$	I.	$1,1$
B.	$\sqrt{1-\left(\dfrac{dy}{dx}\right)^2} = a\left(\dfrac{d^2y}{dx^2}\right)^{1/3}$	II.	$1,2$
C.	$y = \dfrac{dy}{dx} + \dfrac{5}{(dy/dx)}$	III.	$2,1$
D.	$\sqrt{1-y^2}\,dx + \sqrt{1-x^2}\,dy = 0$	IV.	$2,2$

Choose the correct answer from the options given below:

A-IV, B-III, C-II, D-I

A-IV, B-III, C-I, D-II

A-III, B-IV, C-I, D-II

A-III, B-IV, C-II, D-I

Correct Answer

Option 4

Correct Answer

Explanation

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