CUET Mathematics - Calculus - Differential Equations | Free Questions - Page 3

Q1:

2026: 21 May Shift 1

Calculus

>

Differential Equations

Easy

Common

Solution of differential equation $\frac{dy}{dx} = \frac{1+x^2}{1+y^2}$ is

$x^3 - y^3 + 3(x+y) + C = 0$ : where $C$ is an arbitrary constant

$x^3 + y^3 + 3(x-y) + C = 0$ : where $C$ is an arbitrary constant

$x^3 - y^3 + 3(x-y) + C = 0$ : where $C$ is an arbitrary constant

$x^3 + y^3 + 3(x+y) + C = 0$ : where $C$ is an arbitrary constant

Correct Answer

Option 3

Correct Answer

Explanation

Q2:

2026: 21 May Shift 1

Calculus

>

Differential Equations

Medium

Common

Match List-I with List-II

List-I (Differential Equation)	List-II (Degree)
(A) $2(y'')^2 + 3y = 6$	(I) 4
(B) $2y + 5(y')^4 = 7$	(II) 1
(C) $\log x + (y'')^3 = e^x$	(III) 2
(D) $\sqrt{1-2y'} = 5(y'')^{\frac{1}{4}}$	(IV) 3

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 4

Correct Answer

Explanation

Q3:

2026: 21 May Shift 1

Calculus

>

Differential Equations

Medium

Core

The solution of the differential equation $e^{\frac{dy}{dx}} = x+1$ ; $y(0)=5$ is

$y = x\log_e x - x + 5$

$y = x\log_e(1+x) - x$

$y = 2\log_e(1+x) + x$

$y = (1+x)\log_e(1+x) - x + 5$

Correct Answer

Option 4

Correct Answer

Explanation

Q4:

2026: 21 May Shift 1

Calculus

>

Differential Equations

Medium

Core

General solution of the Differential equation $\frac{dy}{dx} + ay = e^{mx}$ is

$(a+m)y = e^{mx} + Ce^{-ax}$ : where $C$ is an arbitrary constant

$y = e^{mx} + Ce^{-ax}$ : where $C$ is an arbitrary constant

$(a+m)y = e^{mx} + C$ : where $C$ is an arbitrary constant

$y.e^{ax} = me^{mx} + C$ : where $C$ is an arbitrary constant

Correct Answer

Option 1

Correct Answer

Explanation

Q5:

2026: 21 May Shift 1

Calculus

>

Differential Equations

Medium

Core

For the differential equation $\frac{\left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} = k, k \neq 0$ , which one of the following is TRUE ?

Non-linear differential equation of order 2 and degree 2

linear differential equation of order 2 and degree 2

Non-linear differential equation of order 2 and degree 3

linear differential equation of order 2 and degree 3

Correct Answer

Option 1

Correct Answer

Explanation

Q6:

2026: 19 May Shift 2

Calculus

>

Differential Equations

Medium

Common

Which of the following statements are TRUE?

A. Order of the differential equation is the order of highest differential appearing in the differential equation.

B. Degree of a differential equation is a real number.

C. Order of a differential equation is whole number.

D. Degree of a differential equation is the degree of the highest order derivative, when differential coefficients are free from radicals and fractions.

Choose the correct answer from the options given below:

A and D only

B and C only

A, B, C and D

B, C and D only

Correct Answer

Option 3

Correct Answer

Explanation

Q7:

2026: 19 May Shift 2

Calculus

>

Differential Equations

Easy

Common

Solution of differential equation $\frac{dy}{dx} = e^{x-y} + 3x^2 e^{-y}$ is:

(where C is an arbitrary constant)

$e^y + e^x = x^3 + C$

$e^y - e^x = x^3 + C$

$e^{x-y} = x^3 + C$

$e^x - e^y = 3x^3 + C$

Correct Answer

Option 2

Correct Answer

Explanation

Q8:

2026: 19 May Shift 2

Calculus

>

Differential Equations

Easy

Core

The equation of the curve that passes through $(-1, 1)$ and the slope of line touching to the curve at any point $(x, y)$ is $\frac{3x^2}{y}$ , is:

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

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

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

$y^2 = 2x^3 + 6$

Correct Answer

Option 1

Correct Answer

Explanation

Q9:

2026: 19 May Shift 2

Calculus

>

Differential Equations

Medium

Core

The general solution of the given differential equation $dy + (2y - e^{-2x})dx = 0$ , is:

[where c is an arbitrary constant]

$2e^{2x}y = x + c$

$e^{2x} + y = x + c$

$e^{-2x}y = x + c$

$e^{2x}y = x + c$

Correct Answer

Option 4

Correct Answer

Explanation

Q10:

2026: 19 May Shift 2

Calculus

>

Differential Equations

Medium

Applied

If m and n are the order and degree of the differential equation $9\frac{d^3y}{dx^3} = \left\{1 + \left(\frac{dy}{dx}\right)^2\right\}^{5/2}$ , then the value of $2m+3n$ is equal to

10

12

15

7

Correct Answer

Option 2

Correct Answer

Explanation

Q11:

2026: 19 May Shift 1

Calculus

>

Differential Equations

Easy

Common

The general solution of the given differential equation $\frac{dy}{dx} = \frac{3e^{2x}+3e^{4x}}{e^{x}+e^{-x}}$ (where C is an arbitrary constant)

$y = 3e^{3x} + C$

$y = \frac{1}{3}e^{3x} + C$

$y = 9e^{3x} + C$

$y = e^{3x} + C$

Correct Answer

Option 4

Correct Answer

Explanation

Q12:

2026: 19 May Shift 1

Calculus

>

Differential Equations

Easy

Common

The solution of the initial value problem $x\frac{dy}{dx} = 2, y(1) = 2$ is

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

$y = 2(\log|x| + 2)$

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

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

Correct Answer

Option 1

Correct Answer

Explanation

Q13:

2026: 19 May Shift 1

Calculus

>

Differential Equations

Medium

Common

Match List-I with List-II

List-I	List-II
(A) $y + \frac{dy}{dx} = \frac{1}{4}\int y\,dx$	(I) Order 1, Degree 1
(B) $y = \frac{dy}{dx} + \frac{C}{dy/dx}$	(II) Order 2, Degree 2
(C) $(xy^2+x)dx + (y-x^2)dy = 0$	(III) Order 2, Degree 1
(D) $\left[\left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{3/2}/\frac{d^2y}{dx^2}\right] = K$	(IV) Order 1, Degree 2

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 2

Correct Answer

Explanation

Q14:

2026: 19 May Shift 1

Calculus

>

Differential Equations

Easy

Core

Match List-I with List-II

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

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 1

Correct Answer

Explanation

Q15:

2026: 19 May Shift 1

Calculus

>

Differential Equations

Easy

Core

The sum of the order and degree of the differential equation: $\left(x + \frac{dy}{dx}\right)^2 = \frac{dy}{dx} + 1$ is

1

3

2

4

Correct Answer

Option 2

Correct Answer

Explanation

Q16:

2026: 19 May Shift 1

Calculus

>

Differential Equations

Medium

Core

The general solution of the differential equation $\frac{dy}{dx} + y\cot x = 2\cos x$ is (Where C is an arbitrary constant)

$(y+\sin x)\sin x = C$

$(y+\cos x)\sin x = C$

$(y-\sin x)\sin x = C$

$(y-\cos x)\sin x = C$

Correct Answer

Option 3

Correct Answer

Explanation

Q17:

2026: 19 May Shift 1

Calculus

>

Differential Equations

Medium

Applied

Match List-I with List-II

List-I (Differential equation)	List-II (The sum of degree and order of the differential equation)
(A) $x\frac{d^2y}{dx^2} = \left(1+\left(\frac{dy}{dx}\right)^2\right)^{\frac{1}{3}}$	(I) 2
(B) $x\frac{d^2y}{dx^2} = \left(2-\frac{dy}{dx}\right)^{\frac{1}{7}}$	(II) 9
(C) $3+\frac{dy}{dx} = y+\frac{d^3y}{dx^3}$	(III) 4
(D) $\frac{dy}{dx} = x+3$	(IV) 5

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 4

Correct Answer

Explanation

Q18:

2026: 18 May Shift 2

Calculus

>

Differential Equations

Easy

Common

Match the LIST-I with LIST-II

LIST-I Differential equation	LIST-II Order and degree
A. $y'' = [1+(y')^2]^{3/2}$	I. 3, 4
B. $y''' + y' + y = 2x$	II. 2, 2
C. $2(y'')^3 - (y')^4 + y = 0$	III. 3, 1
D. $(y''')^4 + (y'')^3 + y' + y = 0$	IV. 2, 3

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 1

Correct Answer

Explanation

Q19:

2026: 18 May Shift 2

Calculus

>

Differential Equations

Medium

Common

The solution of the differential equation $9+y^2\dfrac{dy}{dx}=x^2$ is :

[Where $C$ is an arbitrary constant]

$9y+\dfrac{y^3}{3}=\dfrac{x^3}{3}+C$

$x^3-y^3=27x+C$

$x^3-y^3=27y+C$

$9y+\dfrac{x^3}{3}=\dfrac{y^3}{3}+C$

Correct Answer

Option 2

Correct Answer

Explanation

Q20:

2026: 18 May Shift 2

Calculus

>

Differential Equations

Medium

Core

The integrating factor of the differential equation $y\log_e(10y).\dfrac{dx}{dy}+x=\dfrac{1}{[\log_e(10y)]^2}$ , is

$\log_e y$

$\log_e(10y)$

$\log_e 10$

$1+\log_e y$

Correct Answer

Option 2

Correct Answer

Explanation

Q21:

2026: 18 May Shift 2

Calculus

>

Differential Equations

Hard

Core

The general solution of the differential equation $\tan\left(\dfrac{(1+x^2)y'+2xy}{2x}\right)=1+x^2$ , is

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

(where C is an arbitrary constant)

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

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

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

$y=\tan^{-1}(1+x^2)-\dfrac{1}{2(1+x^2)}\log|x^4+2x^2+2|+C$

Correct Answer

Option 4

Correct Answer

Explanation

Q22:

2026: 18 May Shift 2

Calculus

>

Differential Equations

Easy

Applied

The order and degree of the differential equation $\left[2+\left(\dfrac{dy}{dx}\right)\right]^{5/2}=7\dfrac{d^3y}{dx^3}$ is

Order = 1, degree = 5

Order = 5, degree = 1

Order = 3, degree = 2

Order = 2, degree = 3

Correct Answer

Option 3

Correct Answer

Explanation

Q23:

2026: 18 May Shift 1

Calculus

>

Differential Equations

Easy

Common

Which of the following differential equations are of order 2 and degree 1?

(A). $\frac{d}{dx}\left(\frac{dy}{dx}\right) + x\frac{dy}{dx} = y$

(B). $\frac{d^2y}{dx^2} + x = \sqrt{\frac{dy}{dx} - y}$

(C). $\left(\frac{dy}{dx}\right)^2 + \frac{d^2y}{dx^2} = \log x$

(D). $\left(\frac{dy}{dx}\right)^2 + \frac{dy}{{dx}} = 3$

Choose the correct answer from the options given below:

(A) and (C) only

(A) and (B) only

(C) and (D) only

(B) and (C) only

Correct Answer

Option 1

Correct Answer

Explanation

Q24:

2026: 18 May Shift 1

Calculus

>

Differential Equations

Easy

Common

The solution of the differential equation $x\frac{dy}{dx} = 2x^2 + 1$ ; $x \neq 0$ with $y = 1$ when $x = 1$ , is

$xy = \log_e|x|$

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

$y = x^2 - \log_e|x|$

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

Correct Answer

Option 4

Correct Answer

Explanation

Q25:

2026: 18 May Shift 1

Calculus

>

Differential Equations

Medium

Common

Solution of differential equation $\frac{dy}{dx} = e^{x+y} - xe^y$ is

$e^x + e^y - \frac{x^2}{2} = C$ , where C is an arbitrary constant

$e^x + e^{-y} - \frac{x^2}{2} = C$ , where C is an arbitrary constant

$e^x + e^y + \frac{x^2}{2} = C$ , where C is an arbitrary constant

$e^x - e^{-y} - \frac{x^2}{2} = C$ , where C is an arbitrary constant

Correct Answer

Option 2

Correct Answer

Explanation

Q26:

2026: 18 May Shift 1

Calculus

>

Differential Equations

Easy

Core

Match List-I with List-II

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

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 3

Correct Answer

Explanation

Q27:

2026: 18 May Shift 1

Calculus

>

Differential Equations

Medium

Core

Match List-I with List-II

List-I Differential equation	List-II Curve satisfying the differential equation
(A) $\frac{d^2y}{dx^2}+y=0$	(I) $x^2+y^2=C$ : Where C is an arbitrary constant
(B) $\frac{dy}{dx}+y=\frac{x^2+y}{x}$	(II) $3e^{-4y}+4e^{3x}=C$ : Where C is an arbitrary constant
(C) $\frac{dy}{dx}+\frac{x}{y}=0$	(III) $y=\sin x+\cos x+C$ : Where C is an arbitrary constant
(D) $\log(dy/dx)=3x+4y$	(IV) $y=x+Cxe^{-x}$ : Where C is an arbitrary constant

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 2

Correct Answer

Explanation

Q28:

2026: 18 May Shift 1

Calculus

>

Differential Equations

Medium

Applied

The differential equation satisfied by the equation $y = ae^{bx}$ , where $a$ and $b$ are arbitrary constants, is

$\frac{d^2y}{dx^2} = y\left(\frac{dy}{dx}\right)^2$

$y\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2$

$y\frac{d^2y}{dx^2} = \frac{dy}{dx}$

$\frac{d^2y}{dx^2} = y\frac{dy}{dx}$

Correct Answer

Option 2

Correct Answer

Explanation

Q29:

2026: 18 May Shift 1

Calculus

>

Differential Equations

Medium

Applied

Match List-I with List-II

List-I	List-II
Differential equation	Order and degree
(A) $y''' + x^2(y'')^3 + y = 0$	(I) $3, 2$
(B) $y''' + y^2 + e^{y'} = 0$	(II) $3, 3$
(C) $(y''')^2 + x^2(y'')^3 + xy = 0$	(III) $3, 1$
(D) $(y''')^3 + x(y'')^2 + y = 0$	(IV) $3, \text{Not defined}$

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 2

Correct Answer

Explanation

Q30:

2026: 15 May Shift 1

Calculus

>

Differential Equations

Easy

Common

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

$2^x + 2^{-y} = c$ where $c$ is an arbitrary constant

$2^x + 2^y = c$ where $c$ is an arbitrary constant

$2^x - 2^{-y} = c$ where $c$ is an arbitrary constant

$2^x \times 2^y = c$ where $c$ is an arbitrary constant

Correct Answer

Option 1

Correct Answer

Explanation

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