Match List - I with List - II. Match the integrating factors :, List - I (Differential Equation), List - II (Integrating factor), (A) dy/dx + 3y = e^(-2x), (I) 1/x, (B) xdy/dx + y = 3x^2, (II) e^(-x), (C) xdy/dx - y = 3x^2, (III) x, (D) dy/dx - y = x, (IV) e^3x, Choose the correct answer from the options given below : Option 1: (A)-(IV), (B)-(III), (C)-(I), (D)-(II) Option 2: (A)-(III), (B)-(IV), (C)-(I), (D)-(II) Option 3: (A)-(II), (B)-(III), (C)-(IV), (D)-(I) Option 4: (A)-(I), (B)-(II), (C)-(III), (D)-(IV)

Answer: (A)-(IV), (B)-(III), (C)-(I), (D)-(II). Solution: For form dy/dx + Py = Q, IF = e^( P dx). (A) P = 3 IF = e^3x (IV). (B) dy/dx + y/x = 3x, IF = x (III). (C) dy/dx - y/x = 3x, IF = 1/x (I). (D) P = -1 IF = e^(-x) (II).

Calculus

Differential Equations

Medium

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Match List - I with List - II. Match the integrating factors :

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

Choose the correct answer from the options given below :

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

(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 Option: 1

For form $\frac{dy}{dx} + Py = Q$ , IF = $e^{\int P\,dx}$ .

(A) $P = 3 \Rightarrow$ IF $= e^{3x}$ (IV).

(B) $\frac{dy}{dx} + \frac{y}{x} = 3x$ , IF $= x$ (III).

(D) $P = -1 \Rightarrow$ IF $= e^{-x}$ (II).

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:

2026: 14 May Shift 2

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:

2023: 22 May Shift 3

Integerating factor of $(x \log_e x) \frac{dy}{dx} + y = 2 \log_e x$ is :

2025: 15 May Shift 2

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

2025: 3 June Shift 2

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

2025: 26 May Shift 2

The solution of the differential equation $xdy - ydx = 0$ represents

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