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

Answer: (A) - (III), (B) - (I), (C) - (IV), (D) - (II). Solution: The integrating factor of a linear equation is e^( P d(variable)). (A) P=2y e^(2 y)=y^2, (III). (B) P=1 e^x, (I). (C) P=-12 e^(-x/2), (IV). (D) P=-1/2y e^(-12 y)=1/( y), (II).

Calculus

Differential Equations

Medium

core

Match List-I with List-II

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

Choose the correct answer from the options given below:

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

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

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

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

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

✅ Correct Option: 1

The integrating factor of a linear equation is $e^{\int P\,d(\text{variable})}$ .

(A) $P=\frac2y\Rightarrow e^{2\log y}=y^2$ , (III).

(B) $P=1\Rightarrow e^x$ , (I).

(D) $P=-\frac{1}{2y}\Rightarrow e^{-\frac12\log y}=\frac{1}{\sqrt y}$ , (II).

2022: 6 Aug Shift 2

If the order and the degree of the differential equation $\left(\frac{dy}{dx}\right)^{\frac{1}{2}} = \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{5}}$ are O and S respectively, then $S - O$ is equal to

2025: 21 May Shift 2

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

2026: 30 May Shift 1

Let $y(x)$ be the solution curve of the differential equation

$\frac{dy}{dx} + 2y^2 = 0, y(1) = 1$ , then $y\left(\frac{3}{4}\right)$ is equal to:

2025: 21 May Shift 1

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:

2025: 15 May Shift 2

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

2026: 22 May Shift 2

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:

Solution

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