CUET Mathematics - Calculus - Differential Equations | Free Questions - Page 6

Q1:

2025: 22 May Shift 1

Calculus

>

Differential Equations

Medium

Applied

The differential equation of the family of curves $y = Ae^{3x} + Be^{-3 x}$ , where $a$ and $\beta$ are arbitrary constants, is

$\frac{d^2y}{dx^2} + 9y = 0$

$\frac{d^2y}{dx^2} + 3y = 0$

$\frac{d^2y}{dx^2} = 9y$

$\frac{d^2y}{dx^2} = 3y$

Correct Answer

Option 3

Correct Answer

Explanation

Q2:

2025: 21 May Shift 2

Calculus

>

Differential Equations

Medium

Common

Match List-I with List-II

List-I	List-II
(A) Degree of the differential equation $\frac{d^2y}{dx^2} = e^{dy/dx}$ is	(I) 2
(B) Order of the differential equation $(\frac{dy}{dx})^2 + \frac{d^3y}{dx^3} = 0$ is	(II) not defined
(C) Degree of the differential equation $\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 - 5x^2 = 0$	(III) 3
(D) If p is the order and q is the degree of the differential equation $\frac{dy}{dx} + 3y = e^x$ , then p + q is	(IV) 1

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 3

Correct Answer

Explanation

Q3:

2025: 21 May Shift 2

Calculus

>

Differential Equations

Medium

Common

The solution of the differential equation $(x + 1)\frac{dy}{dx} + 1 - 2e^{-y} = 0$ , $y(0) = 0$ is

$|(x - 1)(e^y - 2)| = 1$

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

$|(x + 1)(e^y + 2)| = 1$

$|(x - 2)(e^y - 2)| = 1$

Correct Answer

Option 2

Correct Answer

Explanation

Q4:

2025: 21 May Shift 2

Calculus

>

Differential Equations

Medium

Core

Match List-I with List-II

List-I	List-II
(A) Integrating factor of $xdy - (y + x^2)dx = 0$	(I) $x^2$
(B) Integrating factor of $xdy + (2y + x^2)dx = 0$	(II) $x^3$
(C) Integrating factor of $(3y - x^2)dx + xdy = 0$	(III) $x$
(D) Integrating factor of $(y + 3x^2)dx + xdy = 0$	(IV) $\frac{1}{x}$

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 1

Correct Answer

Explanation

Q5:

2025: 21 May Shift 2

Calculus

>

Differential Equations

Medium

Core

The solution of the differential equation $\frac{dy}{dx} = \frac{ax + c}{by + d}$ represents a circle when

$a = -b$

$a = b$

$a = -2b$

$a = 3b$

Correct Answer

Option 1

Correct Answer

Explanation

Q6:

2025: 21 May Shift 1

Calculus

>

Differential Equations

Medium

Common

Match List-I with List-II

List-I	List-II
Differential Equation	General solution
---	---
(A) $\dfrac{dy}{dx} = \dfrac{y}{x}; x \neq 0$	(I) $y = cx; c \text{ is an arbitrary constant}$
(B) $x dx - y dy = 0; y \neq 0, x \neq 0$	(II) $x^2 - y^2 = c; c \text{ is an arbitrary constant}$
(C) $\dfrac{(x^2 - 1)}{y^2 + 1}\dfrac{dx}{dy} = 1$	(III) $2x + 3y = c; c \text{ is an arbitrary constant}$
(D) $2 dx + 3 dy = 0$	(IV) $(x^3-y^3) = c + 3(x+y); c \text{ is an arbitrary constant}$

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 4

Correct Answer

Explanation

Q7:

2025: 21 May Shift 1

Calculus

>

Differential Equations

Easy

Common

The number of arbitrary constants in the general solution of a differential equation with degree 1 and order 3, is

1

2

3

4

Correct Answer

Option 3

Correct Answer

Explanation

Q8:

2025: 21 May Shift 1

Calculus

>

Differential Equations

Easy

Core

A integrating factor of the differential equation $\frac{dy}{dx} + \frac{y}{x} = \frac{1}{x^2}$ , $(x > 0)$ is equal to

x

-x

$\frac{1}{x}$

$-\frac{1}{x}$

Correct Answer

Option 1

Correct Answer

Explanation

Q9:

2025: 21 May Shift 1

Calculus

>

Differential Equations

Medium

Core

The general solution of the differential equation $x(1 + y^2)dx + y(1 + x^2)dy = 0$ is

$(1 + y^2)(1 + x^2) = C$ ; where C is an arbitrary constant

$(1 - y^2)(1 - x^2) = C$ ; where C is an arbitrary constant

$(1 - y^2)(1 + x^2) = C$ ; where C is an arbitrary constant

$(1 + y^2)(1 - x^2) = C$ ; where C is an arbitrary constant

Correct Answer

Option 1

Correct Answer

Explanation

Q10:

2025: 21 May Shift 1

Calculus

>

Differential Equations

Medium

Applied

The differential equation representing the curve $y = e^{2x}(a + bx)$ , where a, b are arbitrary constants is

$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 4y = 0$

$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} - 4y = 0$

$\frac{d^2y}{dx^2} - 4\frac{dy}{dx} + 4y = 0$

$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} - 4y = 0$

Correct Answer

Option 3

Correct Answer

Explanation

Q11:

2025: 19 May Shift 1

Calculus

>

Differential Equations

Hard

Common

Match List-I with List-II

List-I	List-II
Differential Equation	Order and degree of differential equation
(A) $\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^2 = e^{\frac{dy}{dx}} + 1$	(I) Order = 1, Degree = 2
(B) $\left(\frac{d^2y}{dx^2}\right)^2 + 4\left(\frac{dy}{dx}\right)^3 = e^y - 1$	(II) Order = 2, Degree = 1
(C) $3\left(\frac{dy}{dx}\right) + 4y + e^y = \frac{dx}{dy}$	(III) Order = 2, Degree = 2
(D) $\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right) = \left(e^y + \frac{dy}{dx}\right)^2$	(IV) Order = 2, Degree = Not defined

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

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

Correct Answer

Option 3

Correct Answer

Explanation

Q12:

2025: 19 May Shift 1

Calculus

>

Differential Equations

Medium

Common

The general solution of the differential equation $\frac{dy}{dx} = e^{ax+by}$ is: (Here C is an arbitrary constant)

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

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

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

$e^{ax} + e^{by} = C$

Correct Answer

Option 3

Correct Answer

Explanation

Q13:

2025: 19 May Shift 1

Calculus

>

Differential Equations

Hard

Core

Solution of the differential equation

$\frac{dy}{dx} = \sqrt{1 + x^2 + y^2 + x^2y^2}$ is : (Here $C$ is an arbitrary constant)

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

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

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

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

Correct Answer

Option 3

Correct Answer

Explanation

Q14:

2025: 19 May Shift 1

Calculus

>

Differential Equations

Hard

Core

Match List-I with List-II

List-I	List-II
Differential Equation	Integrating Factor
(A) $y dx + (x - y^3)dy = 0$	(I) $e^{-x}$
(B) $x\frac{dy}{dx} + y = x^2$	(II) $\frac{1}{x}$
(C) $\frac{dy}{dx} - y = e^x$	(III) $y$
(D) $x dy - y dx = x^3 dx$	(IV) $x$

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 3

Correct Answer

Explanation

Q15:

2025: 19 May Shift 1

Calculus

>

Differential Equations

Medium

Applied

If the slope of the tangent to the curve $y = y(x)$ at any point $(x, y)$ is $\frac{2x}{y^2}$ and the curve passes through the point $\left(\frac{1}{\sqrt3}, 1\right)$ , then equation of curve is

$y^2 = 3x^3$

$y^3 = 3x^2$

$y^4 = 3x^2$

$y = 3x^2$

Correct Answer

Option 2

Correct Answer

Explanation

Q16:

2025: 16 May Shift 1

Calculus

>

Differential Equations

Medium

Common

The degree of the differential equation $\left(2 + \left(\frac{dy}{dx}\right)^2\right)^{\frac{3}{2}} = a^2 \frac{d^2y}{dx^2}$ is:

6

1

3

2

Correct Answer

Option 4

Correct Answer

Explanation

Q17:

2025: 16 May Shift 1

Calculus

>

Differential Equations

Medium

Common

The particular solution of the differential equation $\log\left(\frac{dy}{dx}\right)= 3x + 4y$ satisfying $y = 0$ when $x = 0$ is:

$3e³ˣ + 4e⁻⁴ʸ + 7 = 0$

$3e³ˣ - 4e⁻⁴ʸ - 7 = 0$

$4e³ˣ + 3e⁻⁴ʸ - 7 = 0$

$4e³ˣ - 3e⁻⁴ʸ + 7 = 0$

Correct Answer

Option 3

Correct Answer

Explanation

Q18:

2025: 16 May Shift 1

Calculus

>

Differential Equations

Hard

Core

Given differential equation, (1 + y²)dx = (tan⁻¹y - x)dy, then which of the following is/are true?

(A) Integrating factor = tan⁻¹x

(B) Integrating factor = tan⁻¹y

(C) Integrating factor = $e^{tan⁻¹y}$

(D) Degree = 1

Choose the correct answer from the options given below:

(B) and (D) only

(A) only

(C) only

(C) and (D) only

Correct Answer

Option 4

Correct Answer

Explanation

Q19:

2025: 16 May Shift 1

Calculus

>

Differential Equations

Medium

Core

The general solution of the differential equation $\frac{dy}{dx} + y \tan x = \sec x$

$y\sec x - \tan x = c$ , where c is an arbitary constant

$y\sec x + \tan x = c$ , where c is an arbitary constant

$y \tan x - \sec x = c$ , where c is an arbitary constant

$y \tan x + \sec x = c$ , where c is an arbitary constant

Correct Answer

Option 1

Correct Answer

Explanation

Q20:

2025: 16 May Shift 1

Calculus

>

Differential Equations

Medium

Applied

The particular solution of the differential equation x(1 + y²)dx - y(1 + x²)dy = 0, y(0) = 1, is:

$xy^2 = x^2 + 1$

$y^2 = 2x^2 + 1$

$y^2 = 2x^2 + xy + 1$

$x^2 = 3y^2 + 2xy + 1$

Correct Answer

Option 2

Correct Answer

Explanation

Q21:

2025: 15 May Shift 2

Calculus

>

Differential Equations

Medium

Common

Match List-I with List-II

List-I	List-II
Differential equation	Order and degree
(A) $(y'')^3 + (y')^4 - 6 = (y''')^2$	(I) Order = 1, Degree = 2
(B) $\sqrt{(y')^2 + 5} = y''$	(II) Order = 2, Degree = 3
(C) $(y')^2 = (2 + y'')^{3/2}$	(III) Order = 2, Degree = 2
(D) $y = xy' + \sqrt{a^2(y')^2 + b^2}$	(IV) Order = 3, Degree = 2

Choose the correct answer from the options given below:

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

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

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

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

Correct Answer

Option 3

Correct Answer

Explanation

Q22:

2025: 15 May Shift 2

Calculus

>

Differential Equations

Medium

Common

Consider the differential equation $xdy = (x + y) dx$ . Which of the following are true?

(A) It is a homogenous differential equation

(B) It is a differential equation of order 2

(C) The general solution of the differential equation contains 2 arbitrary constants

(D) Integrating factor of differential equation is $\frac{1}{x}$

(E) Degree of the differential equation is not defined

Choose the correct answer from the options given below:

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

(A), (D) and (E) only

(A) and (D) only

(B) and (C) only

Correct Answer

Option 3

Correct Answer

Explanation

Q23:

2025: 15 May Shift 2

Calculus

>

Differential Equations

Medium

Core

The general solution of the differential equation $\frac{xdy}{dx} + 4y = x^3, (x \neq 0)$ is:

$y = \frac{x^3}{7} + cx^{-4}$ ; $c$ is arbitrary constant.

$y = \frac{x^3}{7} + cx^{-7}$ ; $c$ is arbitrary constant.

$y = \frac{x^7}{7} + cx^{-4}$ ; $c$ is arbitrary constant.

$y = \frac{x^7}{7} + cx^{-3}$ ; $c$ is arbitrary constant.

Correct Answer

Option 1

Correct Answer

Explanation

Q24:

2025: 15 May Shift 2

Calculus

>

Differential Equations

Medium

Core

Consider the curve which is represented by the differential equation $\frac{dy}{dx} = 1 + x + y + xy$ . If it passes through the point $(0,0)$ , then which of the following is/are true?

(A) it is a straight line.

(B) it is a parabola.

(C) it also passes through the point $(-1, \frac{1}{\sqrt{e}} - 1)$

(D) Its equation is $xy(x + 1)\left(y - \frac{1}{\sqrt{e}} + 1\right) = 0$

Choose the correct answer from the options given below:

(C) and (D) only

(A) only

(B) only

(C) only

Correct Answer

Option 4

Correct Answer

Explanation

Q25:

2025: 15 May Shift 2

Calculus

>

Differential Equations

Hard

Applied

The general solution of the differential equation $(x^2 - yx^2)dy + (y^2 + x^2y^2)dx = 0$ is:

$\log_e|y| + \frac{1}{x} + \frac{1}{y} - x = c$ , where $c$ is constant of integration

$\log_e|y| - \frac{1}{x} + \frac{1}{y} + x = c$ , where $c$ is constant of integration

$\log_e|x| - \frac{1}{x} + \frac{1}{y} + x = c$ , where $c$ is constant of integration

$\log_e|x| + \frac{1}{x} + \frac{1}{y} + x = c$ , where $c$ is constant of integration

Correct Answer

Option 1

Correct Answer

Explanation

Q26:

2025: 15 May Shift 1

Calculus

>

Differential Equations

Medium

Common

The general solution of the differential equation $(1 + e^x)dy + ye^x dx = 0$ , where $y > 0$ , is

$y = C(1 + e^x)$ , C is an arbitary constant

$\frac{C}{e^x} = \frac{y}{1 + e^x}$ , C is an arbitary constant

$C = y(1 + e^x)$ , C is an arbitary constant

$y = \frac{Ce^x}{1 + e^x}$ , C is an arbitary constant

Correct Answer

Option 3

Correct Answer

Explanation

Q27:

2025: 15 May Shift 1

Calculus

>

Differential Equations

Medium

Common

Match List-I with List-II

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

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

Q28:

2025: 15 May Shift 1

Calculus

>

Differential Equations

Hard

Core

For $|x| < 1$ , if $x = \cos\left(\frac{1}{a}\log y\right)$ , then

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

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

$(1 - x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} - a^2y = 0$

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

Correct Answer

Option 3

Correct Answer

Explanation

Q29:

2025: 15 May Shift 1

Calculus

>

Differential Equations

Medium

Core

Consider the differential equation $\frac{dy}{dx} + y \tan x = \sec x$ , then which of the following statements are correct?

(A) It is homogeneous

(B) It has $\sec x$ as its integrating factor

(C) It's general solution is $y \sec x = \tan x + c$ , where c is arbitary constant.

(D) It's degree is not defined

Choose the correct answer from the options given below:

(A) and (D) only

(B) and (C) only

(C) and (D) only

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

Correct Answer

Option 2

Correct Answer

Explanation

Q30:

2025: 15 May Shift 1

Calculus

>

Differential Equations

Hard

Core

For $y \neq 0$ , the particular solution of the differential equation $2ye^{x/y}dx + (y - 2xe^{x/y})dy = 0$ at the point (1, 1) is

$\ln|y| + 2(e^{x/y} - e) = 0$

$\ln|y| - 2(e^{x/y} - e) = 0$

$\ln|y| + \frac{1}{2}(e^{x/y} - e) = 0$

$\ln|y| - \frac{1}{2}(e^{x/y} - e) = 0$

Correct Answer

Option 1

Correct Answer

Explanation

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