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

Answer: (A) - (III), (B) - (IV), (C) - (I), (D) - (II). Solution: Reduce each to dy/dx+Py = Q and use IF = e^( P dx). (A) P = -1/x, IF = 1/x. (B) P = -2, IF = e^(-2x). (C) P = 1/x, IF = x. (D) P = 4/x, IF = x^4.

Calculus

Differential Equations

Medium

core

Match List-I with List-II

List-I (Differential equation) List-II (Integrating factor)
(A) $\dfrac{xdy}{dx}-y=x^2e^x$ (I) $x$
(B) $\dfrac{dy}{dx}-2y=e^x$ (II) $x^4$
(C) $\dfrac{xdy}{dx}+y=1$ (III) $\dfrac{1}{x}$
(D) $\dfrac{xdy}{dx}+4y=x^4$ (IV) $e^{-2x}$

Choose the correct answer from the options given below:

List-I (Differential equation)	List-II (Integrating factor)
(A) $\dfrac{xdy}{dx}-y=x^2e^x$	(I) $x$
(B) $\dfrac{dy}{dx}-2y=e^x$	(II) $x^4$
(C) $\dfrac{xdy}{dx}+y=1$	(III) $\dfrac{1}{x}$
(D) $\dfrac{xdy}{dx}+4y=x^4$	(IV) $e^{-2x}$

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

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

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

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

✅ Correct Option: 1

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

(A) $P = -\dfrac{1}{x}$ , $IF = \dfrac{1}{x}$ .

(B) $P = -2$ , $IF = e^{-2x}$ .

(D) $P = \dfrac{4}{x}$ , $IF = x^4$ .

2025: 27 May Shift 1

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

2026: 19 May Shift 1

Match List-I with List-II

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

Choose the correct answer from the options given below:

2025: 3 June Shift 1

The sum of order and degree of the differential equation $(x^2\frac{d^2y}{dx^2})^{3/4} = 5(\frac{dy}{dx})^2 - 3$ is equal to

2026: 22 May Shift 2

Particular solution of the differential equation $dy = e^{2x+y}dx$ , $y(0) = 0$ is:

2026: 19 May Shift 2

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

[where c is an arbitrary constant]

2022: 10 Aug Shift 1

If the order and degree of the differential equation $\sqrt{\frac{d^2y}{dx^2}} = \left(1 + \frac{dy}{dx}\right)^{\frac{1}{3}}$ are $a$ and $b$ respectively, then the value of $a^2 + b^2$ is

Solution

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