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

Answer: (A) - (IV), (B) - (III), (C) - (II), (D) - (I). Solution: Reduce each to dy/dx+Py=Q and use IF=e^( P dx). (A) P=-1x, IF=1x, so (IV). (B) P=1/2x, IF=x^(1/2), so (III). (C) P=1, IF=e^x, so (II). (D) P=-1, IF=e^(-x), so (I).

Calculus

Differential Equations

Medium

core

Match List-I with List-II

List-I: Linear differential equation List-II: Integrating factor
(A) $x\frac{dy}{dx}-y=2x^3$ (I) $e^{-x}$
(B) $2x\frac{dy}{dx}+y=3x^2$ (II) $e^x$
(C) $\frac{dy}{dx}+y=\cos x-\sin x$ (III) $x^{1/2}$
(D) $\frac{dy}{dx}-y=e^x$ (IV) $1/x$

Choose the correct answer from the options given below:

List-I: Linear differential equation	List-II: Integrating factor
(A) $x\frac{dy}{dx}-y=2x^3$	(I) $e^{-x}$
(B) $2x\frac{dy}{dx}+y=3x^2$	(II) $e^x$
(C) $\frac{dy}{dx}+y=\cos x-\sin x$	(III) $x^{1/2}$
(D) $\frac{dy}{dx}-y=e^x$	(IV) $1/x$

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

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

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

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

✅ Correct Option: 4

Reduce each to $\frac{dy}{dx}+Py=Q$ and use $IF=e^{\int P dx}$ .

(A) $P=-\frac1x$ , $IF=\frac1x$ , so (IV).

(B) $P=\frac{1}{2x}$ , $IF=x^{1/2}$ , so (III).

(D) $P=-1$ , $IF=e^{-x}$ , so (I).

2026: 30 May Shift 1

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

2026: 26 May Shift 2

General solution of the differential equation $(1+x^2)dy + 2xy\,dx = \cot x\,dx$ is:

(where $C$ is an arbitrary constant)

2022: 7 Aug Shift 2

The general solution of the differential equation $\frac{dy}{dx} = e^{x+y}$ , is :

2025: 3 June Shift 1

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

2025: 13 May Shift 2

The general solution of the differential equation

$\frac{dy}{dx} = xy + x + y + 1$ is

2025: 30 May Shift 1

For the differential equation $ydx - (x + 3y^2)dy = 0$ , which of the following statements are true?

(A) It is a linear differential equation

(B) It is a homogenous differential equation

(D) If $y(0) = 1$ , then its particular solution is $x = 3y^2 - 1$

Choose the correct answer from the options given below:

Solution

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