CUET Mathematics - Calculus - Differential Equations | Free Questions - Page 4

Q1:

2026: 15 May Shift 1

Calculus

>

Differential Equations

Medium

Common

The order and degree of differential equation $y = px + \sqrt{a^2p^2 + b^2}, p = \dfrac{dy}{dx}$ , $a$ and $b$ are constants are:

order = 1, degree = 1

order = 2, degree = 1/2

order = 1, degree not defined

order = 1, degree = 2

Correct Answer

Option 4

Correct Answer

Explanation

Q2:

2026: 15 May Shift 1

Calculus

>

Differential Equations

Medium

Common

If $c$ is an arbitrary constant, then

Match the LIST-I with LIST-II

	LIST-I (Differential Equation)		LIST-II (Solution)
A.	$\dfrac{dy}{dx} = 1 + x + y + xy$	I.	$\log_e \vert y \vert = 2x + \log_e (x-1)^2 + c$
B.	$(x-1)\dfrac{dy}{dx} = 2xy$	II.	$y = cx$
C.	$\dfrac{dy}{dx} = 1 - x + y - xy$	III.	$\log_e \vert 1+y \vert = x + \dfrac{x^2}{2} + c$
D.	$x\,dy = y\,dx$	IV.	$\log_e \vert 1+y \vert = x - \dfrac{x^2}{2} + c$

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 1

Correct Answer

Explanation

Q3:

2026: 15 May Shift 1

Calculus

>

Differential Equations

Easy

Core

If $p$ and $q$ are degree and order of the differential equation $\left(\dfrac{d^2y}{dx^2}\right)^3 - 2\left(\dfrac{dy}{dx}\right)^4 = 6$ respectively. Then the value of $2p+3q$ is

13

12

11

14

Correct Answer

Option 2

Correct Answer

Explanation

Q4:

2026: 15 May Shift 1

Calculus

>

Differential Equations

Medium

Core

The solution of the differential equation $\dfrac{dy}{dx} = \dfrac{x^2+xy+y^2}{x^2}$ for $x>0, y>0$ at $x=1, y=2$ is

$\tan\left(\dfrac{y}{x}\right) = \log_e x + \tan 2$

$\tan^{-1}\left(\dfrac{x}{y}\right) = \log_e x + \tan^{-1}2$

$\tan^{-1}\dfrac{y}{x} = \log_e x + \tan^{-1}2$

$\tan^{-1}\dfrac{y}{x} = \log_e x - \tan^{-1}2$

Correct Answer

Option 3

Correct Answer

Explanation

Q5:

2026: 15 May Shift 1

Calculus

>

Differential Equations

Easy

Applied

Match the LIST-I with LIST-II

	LIST-I (Differential equation)		LIST-II (Sum of order and degree)
A.	$(y'')^3 + xy' + 2y = 0$	I.	$6$
B.	$(y'')^2 + (y')^3 + 4y = 0$	II.	$3$
C.	$y'' + (y')^2 + 5y = 0$	III.	$4$
D.	$(y'')^4 + 3x(y')^2 + y = 0$	IV.	$5$

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 4

Correct Answer

Explanation

Q6:

2026: 14 May Shift 2

Calculus

>

Differential Equations

Easy

Common

The solution of the differential equation $\frac{dy}{dx} = x^2 + x + \frac{1}{x}$ is

$y = x^3 + x^2 - \frac{1}{x^2} + C$ : C is an arbitrary constant

$y = \frac{x^3}{3} + \frac{x^2}{2} + \log_e |x| + C$ : C is an arbitrary constant

$y = x + x^2 + \log_e |x| + C$ : C is an arbitrary constant

$y = x^3 + x^2 + \log_e |x| + C$ : C is an arbitrary constant

Correct Answer

Option 2

Correct Answer

Explanation

Q7:

2026: 14 May Shift 2

Calculus

>

Differential Equations

Easy

Common

Particular solution of the differential equation $dy + 2xy^2 dx = 0$ , given that $y=1$ , when $x=0$ , is

$y = x^2+1$

$y = x^2-1$

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

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

Correct Answer

Option 3

Correct Answer

Explanation

Q8:

2026: 14 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.	$\log\left(\dfrac{dy}{dx}\right) + \dfrac{d^2y}{dx^2} = 0$	I.	Order : 3 and degree : not defined
B.	$\dfrac{d^3y}{dx^3} + y^2 + e^{\frac{dy}{dx}} = 4$	II.	Order : 2 and degree : 2
C.	$\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)^2 + \dfrac{dy}{dx} = \sqrt{x}$	III.	Order : 2 and degree : not defined
D.	$\left(1+\left(\dfrac{dy}{dx}\right)^2\right)^{3/2} = \dfrac{d^2y}{dx^2}$	IV.	Order : 2 and degree : 1

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

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

Correct Answer

Option 1

Correct Answer

Explanation

Q9:

2026: 14 May Shift 2

Calculus

>

Differential Equations

Easy

Core

Match the LIST-I with LIST-II

LIST-I		LIST-II
A.	The degree of the differential equation $\left(\frac{d^3y}{dx^3}\right)^4+\left(\frac{d^2y}{dx^2}\right)^5+\left(\frac{dy}{dx}\right)+y=0$	I.	0
B.	The order of the differential equation $\left(\frac{d^3y}{dx^3}\right)^4+\left(\frac{d^2y}{dx^2}\right)^5+\left(\frac{dy}{dx}\right)+y^2=0$	II.	4
C.	The number of arbitrary constants in the general solution of a differential equation of second order is	III.	3
D.	The number of arbitrary constants in a particular solution of a differential equation of third order is	IV.	2

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 2

Correct Answer

Explanation

Q10:

2026: 14 May Shift 2

Calculus

>

Differential Equations

Medium

Core

The general solution of the differintial equation : $x\frac{dy}{dx}-ay=1(a\neq0)$ is

$y=-\frac{1}{a}+Cx^a$ :Where C is an arbitrary constant

$y=-\frac{1}{a}+Cx^{-a}$ :Where C is an arbitrary constant

$y=-\frac{1}{a}+Cx$ :Where C is an arbitrary constant

$y=-\frac{1}{a}+Cx^{-a-1}$ :Where C is an arbitrary constant

Correct Answer

Option 1

Correct Answer

Explanation

Q11:

2026: 12 May Shift 2

Calculus

>

Differential Equations

Medium

Common

Match List-I with List-II

List-I (Differential Equation)	List-II (Order and degree)
(A) $\left(\dfrac{d^2y}{dx^2}\right)^2 + e^{\frac{dy}{dx}} = 0$	(I) order = 2, degree = 1
(B) $xy\dfrac{d^2y}{dx^2} + x\left(\dfrac{dy}{dx}\right)^2 - y\dfrac{dy}{dx} = 0$	(II) order = 2, degree = not defined
(C) $\left(\dfrac{d^3y}{dx^3}\right)^2 + \log_e\left(\dfrac{dy}{dx}\right) = 4x$	(III) order = 3, degree = 2
(D) $\dfrac{d}{dx}\left(\dfrac{d^2y}{dx^2}\right) + y\dfrac{dy}{dx} = \left(1+\dfrac{dy}{dx}\right)^{1/2}$	(IV) order = 3, degree = not defined

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 3

Correct Answer

Explanation

Q12:

2026: 12 May Shift 2

Calculus

>

Differential Equations

Medium

Common

The solution of the differential equation $\dfrac{dy}{dx} = x^2y^2$ with $y = 1$ at $x = 0$ is

$y(x^3-3)-3=0$

$y(x^3+3)-3=0$

$y(x^3+3)+3=0$

$y(x^3-3)+3=0$

Correct Answer

Option 4

Correct Answer

Explanation

Q13:

2026: 12 May Shift 2

Calculus

>

Differential Equations

Medium

Common

The general solution of the differential equation $\dfrac{dy}{dx} = 1+x+y+xy$ is: (where $C$ is an arbitrary constant)

(A) $1+y = x+\dfrac{x^2}{2}+C$

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

(C) $\log_e|1-y| = x-\dfrac{x^2}{2}+C$

(D) $\log_e|1+y| = x+\dfrac{x^2}{2}+C$

Choose the correct answer from the options given below:

(A) only

(B) only

(B) and (D) only

(B) and (C) only

Correct Answer

Option 3

Correct Answer

Explanation

Q14:

2026: 12 May Shift 2

Calculus

>

Differential Equations

Medium

Core

The general solution of the differential equation $ydx + (x-y^3)dy = 0$ is :

$xy = \dfrac{y^4}{4}+cx$ : $c$ is an arbitrary constant.

$x = \dfrac{y^4}{4}+c$ : $c$ is an arbitrary constant.

$xy = \dfrac{y^4}{4}+c$ : $c$ is an arbitrary constant.

$xy = \dfrac{y^3}{3}+C$ : $C$ is an arbitrary constant.

Correct Answer

Option 3

Correct Answer

Explanation

Q15:

2026: 12 May Shift 2

Calculus

>

Differential Equations

Medium

Core

Which of the following differential equations has $y=x$ as one of the particular solution?

$\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}+xy=x$

$\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+xy=x$

$\dfrac{d^2y}{dx^2}-x^2\dfrac{dy}{dx}+xy=0$

$\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+xy=0$

Correct Answer

Option 3

Correct Answer

Explanation

Q16:

2025: 3 June Shift 2

Calculus

>

Differential Equations

Medium

Common

The general solution of the differential equation $\frac{dy}{dx} = -4xy^2$ is given by

$2x^2 - y = C$ ; C is an arbitrary constant

$2x^2 - \frac{1}{y} = C$ ; C is an arbitrary constant

$2x^2 + \frac{1}{y^2} = C$ ; C is an arbitrary constant

$2x^2 + \frac{1}{y} = C$ ; C is an arbitrary constant

Correct Answer

Option 2

Correct Answer

Explanation

Q17:

2025: 3 June Shift 2

Calculus

>

Differential Equations

Medium

Common

Match List-I with List-II

List-I	List-II
(A) Degree of this differential equation $\frac{d^4y}{dx^4} + 2\log_e\left(\frac{d^3y}{dx^3}\right) = 0$	(I) 1
(B) Order of this differential equation $e^{\left(\frac{dy}{dx}\right)^3} + 3y\left(\frac{d^2y}{dx^2}\right)^3 = 0$	(II) 4
(C) Degree of $\frac{d^4y}{dx^4} + \left(\frac{dy}{dx}\right)^2 = 0$	(III) not defined
(D) Order of the differential equation $2\frac{d^4y}{dx^4} + \left(\frac{d^2y}{dx^2}\right)^5 = 0$	(IV) 2

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

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

Correct Answer

Option 2

Correct Answer

Explanation

Q18:

2025: 3 June Shift 2

Calculus

>

Differential Equations

Medium

Core

For the differential equation $x\frac{dy}{dx} + 2y = x^2\log_e x$

(A) Integrating factor is $2x$

(B) Integrating factor is $x^2$

(C) General Solution is $y = \frac{x^2}{16}(4\log_e|x| - 1) + Cx^{-2}$ Where C is an arbitrary constant.

(D) General Solution is $y = \frac{x^4}{16}(4\log_e|x| - 1) + C$ Where C is an arbitrary constant.

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 3

Correct Answer

Explanation

Q19:

2025: 3 June Shift 2

Calculus

>

Differential Equations

Easy

Core

The number of arbitrary constants in the particular solution of a differential equation of order 4 and degree 3 is

0

1

3

4

Correct Answer

Option 1

Correct Answer

Explanation

Q20:

2025: 3 June Shift 2

Calculus

>

Differential Equations

Medium

Applied

The sum of the order and degree of the differential equation representing the family of curves $y = mx + m^4$ , where m is arbitrary constant, is

3

4

5

2

Correct Answer

Option 3

Correct Answer

Explanation

Q21:

2025: 3 June Shift 1

Calculus

>

Differential Equations

Medium

Common

The particular solution of the differential equation $xdy = (2x^2 + 1)dx, x \neq 0$ , given that $y = 1$ when $x = 1$ is:

$y = x + \log|x|$

$y = 2 + \log|x|$

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

$y = x^2 + \log|x| + 1$

Correct Answer

Option 3

Correct Answer

Explanation

Q22:

2025: 3 June Shift 1

Calculus

>

Differential Equations

Medium

Common

If m and n are respectively the order and degree of the differential equation $(\frac{d^2y}{dx^2})^{2} + (\frac{dy}{dx})^3 + y= 4x$ , then the value of $m + n$ is:

2

3

4

5

Correct Answer

Option 3

Correct Answer

Explanation

Q23:

2025: 3 June Shift 1

Calculus

>

Differential Equations

Medium

Core

Consider the differential equation $xdy = (y + 2x^3)dx$ . Then which of the following are TRUE?

(A) It is a homogeneous differential equation.

(B) Product of the order and degree of the differential equation in one.

(C) Integrating factor is x.

(D) General solution of the differential equation is $y = x^3 + Cx$ , where C is an arbitary constant.

Choose the correct answer from the options given below:

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

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

(B) and (D) only

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

Correct Answer

Option 3

Correct Answer

Explanation

Q24:

2025: 3 June Shift 1

Calculus

>

Differential Equations

Medium

Core

The sum of order and degree of the differential equation $(x^2\frac{d^2y}{dx^2})^{3/4} = 5(\frac{dy}{dx})^2 - 3$ is equal to

1

9

5

7

Correct Answer

Option 3

Correct Answer

Explanation

Q25:

2025: 3 June Shift 1

Calculus

>

Differential Equations

Medium

Applied

Match List-I with List-II

List-I	List-II
Differential Equation	Sum of order and degree
(A) $\frac{d^2y}{dx^2} + \frac{dy}{dx} + 3y = \sin x$	(I) 2
(B) $\frac{dy}{dx} = \sin(x + y)$	(II) 3
(C) $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2y}{dx^2}$	(III) 4
(D) $x^2(\frac{d^2y}{dx^2})^3 + y(\frac{dy}{dx})^4 + y^5 = 0$	(IV) 5

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 2

Correct Answer

Explanation

Q26:

2025: 3 June Shift 1

Calculus

>

Differential Equations

Medium

Applied

Solution of the differential equation $y\log_e y dx - x dy = 0$ is (Where c is an arbitrary constant)

$|y| = |c\log_e(xy)|$

$|x| = |cy|$

$|x| = |c\log_e y|$

$|y| = |c\log_e x|$

Correct Answer

Option 3

Correct Answer

Explanation

Q27:

2025: 2 June Shift 1

Calculus

>

Differential Equations

Medium

Common

The solution of the differential equation $ydx + (x - y^2)dy = 0$ is

$3xy + x^3 = C$ , C is an arbitrary constant

$3xy + y^3 = C$ , C is an arbitrary constant

$3xy - y^3 = C$ , C is arbitrary constant

$3xy - x^3 = C$ , C is arbitary constant

Correct Answer

Option 3

Correct Answer

Explanation

Q28:

2025: 2 June Shift 1

Calculus

>

Differential Equations

Medium

Common

Match List-I with List-II

List-I	List-II
(A) The degree of differential equation $\frac{d^3y}{dx^3} = e^{\frac{dx}{dy}}$	(I) 2
(B) The order of differential equation $\left(\frac{dy}{dx}\right)^2 + \frac{d^3y}{dx^3} = 0$	(II) 4
(C) The sum of order and degree of differential equation $\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right) + \left(\frac{dy}{dx}\right)^5 = x$	(III) not defined
(D) The number of arbitrary constants in the general solution of a differential equation of order 2	(IV) 3

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 3

Correct Answer

Explanation

Q29:

2025: 2 June Shift 1

Calculus

>

Differential Equations

Medium

Core

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

$y = \frac{x}{2}\log_e(x^2 + y^2) + C$ : C is an arbitary constant

$y = \sqrt{x^2 + y^2} + C$ : C is an arbitrary constant

$y = \frac{1}{2}\log_e(x^2 + y^2) + c$ : C is an arbitrary constant

$\tan^{-1}\frac{y}{x} = \frac{1}{2}\log_e(x^2 + y^2) + C$ : C is an arbitrary constant

Correct Answer

Option 4

Correct Answer

Explanation

Q30:

2025: 2 June Shift 1

Calculus

>

Differential Equations

Medium

Core

The integrating factor of the differential equation $\frac{dy}{dx} = x + xy$ is

$e^{-x/2}$

$e^{-x^2/2}$

$e^{x^2/2}$

$e^{x/2}$

Correct Answer

Option 2

Correct Answer

Explanation

CUET Mathematics: CalculusDifferential Equations. Free, no login required.

Q1:

Q2:

Q3:

Q4:

Q5:

Q6:

Q7:

Q8:

Q9:

Q10:

Q11:

Q12:

Q13:

Q14:

Q15:

Q16:

Q17:

Q18:

Q19:

Q20:

Q21:

Q22:

Q23:

Q24:

Q25:

Q26:

Q27:

Q28:

Q29:

Q30:

CUET Mathematics Past Year Questions (Topic-Wise)