In the context of differential equation Match List I with List II, LIST I, LIST II, A. dy/dx = (1+y^2)/(1+x^2), I. Not a differential equation, B. x^2 dy/dx = x^2 - 2y^2 + xy, II. Linear first order, C. x + y = (x+y), III. Variable separable, D. (x+y)dy/dx = 1, IV. Homogenous, Choose the correct answer from the options given below: Option 1: A-I, B-II, C-III, D-IV Option 2: A-II, B-IV, C-III, D-I Option 3: A-III, B-IV, C-I, D-II Option 4: A-IV, B-I, C-III, D-II

Answer: A-III, B-IV, C-I, D-II. Solution: A: dy/dx = (1+y^2)/(1+x^2) separates as dy/(1+y^2) = dx/(1+x^2), so variable separable (III). B: x^2(dy/dx) = x^2 - 2y^2 + xy has all terms of degree 2, homogeneous (IV). C: x + y = (x+y) contains no derivative, not a differential equation (I). D: (x+y)(dy/dx) = 1 rewrites as dx/dy = x+y, linear first order in x (II).

Calculus

Differential Equations

Medium

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In the context of differential equation

Match List I with List II

LIST I LIST II
A. $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$ I. Not a differential equation
B. $x^2 \frac{dy}{dx} = x^2 - 2y^2 + xy$ II. Linear first order
C. $\sin x + y = \cos(x+y)$ III. Variable separable
D. $(x+y)\frac{dy}{dx} = 1$ IV. Homogenous

Choose the correct answer from the options given below:

LIST I	LIST II
A. $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$	I. Not a differential equation
B. $x^2 \frac{dy}{dx} = x^2 - 2y^2 + xy$	II. Linear first order
C. $\sin x + y = \cos(x+y)$	III. Variable separable
D. $(x+y)\frac{dy}{dx} = 1$	IV. Homogenous

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

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

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

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

✅ Correct Option: 3

A: $dy/dx = (1+y^2)/(1+x^2)$ separates as $dy/(1+y^2) = dx/(1+x^2)$ , so variable separable (III). B: $x^2(dy/dx) = x^2 - 2y^2 + xy$ has all terms of degree 2, homogeneous (IV). C: $\sin x + y = \cos(x+y)$ contains no derivative, not a differential equation (I). D: $(x+y)(dy/dx) = 1$ rewrites as $dx/dy = x+y$ , linear first order in $x$ (II).

2026: 29 May Shift 1

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:

2025: 16 May Shift 1

The particular solution of the differential equation x(1 + y²)dx - y(1 + x²)dy = 0, y(0) = 1, is:

2025: 3 June Shift 1

The particular solution of the differential equation $xdy = (2x^2 + 1)dx, x \neq 0$ , given that $y = 1$ when $x = 1$ is:

2023: 15 June Shift 2

The solution of the differential equation $\frac{dy}{dx} = \frac{6}{x^2}$ ; $y(1) = 3$ is :

2026: 30 May Shift 2

The particular solution of the differential equation $\sec^2x\tan y\,dx - \sec^2y\tan x\,dy = 0$ , $y=\frac{\pi}{4}$ when $x=\frac{\pi}{4}$ , is

2026: 29 May Shift 1

Which of the following are correct?

(A) The order of the differential equation $\left(\dfrac{dy}{dx}\right)^2+3y^2-5y=0$ is 2.

(B) The degree of the differential equation $x\dfrac{d^2y}{dx^2}=\left(3+2\left(\dfrac{dy}{dx}\right)^2\right)^3$ is 3.

(D) The general solution of an ordinary differential equation of $n^{th}$ order contains $n$ independent arbitrary constants.

Choose the correct answer from the options given below:

Solution

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