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 = x + C, (I) xdy/dx - y = 0, (B) y = e^x + C, (II) xdy/dx - 1 = 0, (C) y = cx, (III) x^2dy/dx + 1 = 0, (D) y = 1/x + C, (IV) d^2y/dx^2 - dy/dx = 0, Choose the correct answer from the options given below: Option 1: (A) - (III), (B) - (II), (C) - (I), (D) - (IV) Option 2: (A) - (II), (B) - (IV), (C) - (I), (D) - (III) Option 3: (A) - (II), (B) - (IV), (C) - (III), (D) - (I) Option 4: (A) - (III), (B) - (I), (C) - (IV), (D) - (II)

Answer: (A) - (II), (B) - (IV), (C) - (I), (D) - (III). Solution: Differentiate each curve and eliminate the constant. (A) y= x+C y'=1/x xy'-1=0, (II). (B) y=e^x+C y'=y''=e^x y''-y'=0, (IV). (C) y=cx y'=c=y/x xy'-y=0, (I). (D) y=1/x+C y'=-1/x^2 x^2y'+1=0, (III).

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:

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

(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 Option: 2

Differentiate each curve and eliminate the constant.

(A) $y=\log x+C\Rightarrow y'=1/x\Rightarrow xy'-1=0$ , (II).

(B) $y=e^x+C\Rightarrow y'=y''=e^x\Rightarrow y''-y'=0$ , (IV).

(D) $y=1/x+C\Rightarrow y'=-1/x^2\Rightarrow x^2y'+1=0$ , (III).

2026: 18 May Shift 2

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

2026: 31 May Shift 2

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:

2026: 12 May Shift 2

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:

2023: 22 May Shift 3

The differential equation $y = xp + \sqrt{x^2 p^3 + 4}$ where $p = \frac{dy}{dx}$ is :

(A) of order 1

(B) of degree 1

(D) of degree 3

Choose the correct answer from the options given below :

2022: 16 July Shift 2

The order and the degree of the differential equation $\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 = 2x^2 \log\left(\frac{d^2y}{dx^2}\right)$ are respectively:

2022: 25 May Shift 1

Particular solution of the differential equation $\log\left(\frac{dy}{dx}\right) = x + y$ , given that when $x = 0, y = 0$ is:

Solution

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