Solution of dy/dx = (1+x^2)(1+y^2) is: Option 1: ^(-1) y = x + x^3/3 + c Option 2: ^(-1) y = x - x^3/3 + c Option 3: ^(-1) y = x^2 + x^3/3 + c Option 4: ^(-1) y = x^2 - x^3/3 + c

Answer: ^(-1) y = x + x^3/3 + c. Solution: Separating variables: dy/(1+y^2) = (1+x^2) dx. Integrating both sides: ^(-1) y = x + x^3/3 + c.

Calculus

Differential Equations

Easy

core

Solution of $\frac{dy}{dx} = (1+x^2)(1+y^2)$ is:

$\tan^{-1} y = x + \frac{x^3}{3} + c$

$\tan^{-1} y = x - \frac{x^3}{3} + c$

$\tan^{-1} y = x^2 + \frac{x^3}{3} + c$

$\tan^{-1} y = x^2 - \frac{x^3}{3} + c$

✅ Correct Option: 1

Separating variables: $\frac{dy}{1+y^2} = (1+x^2)\,dx$ . Integrating both sides: $\tan^{-1} y = x + \frac{x^3}{3} + c$ .

2026: 25 May Shift 1

Match the LIST-I with LIST-II

LIST-I Differential equation	LIST-II Integrating factor of the differential equation
A. $\dfrac{dy}{dx} + 2y = xe^{4x}$	I. $\dfrac{1}{x}$
B. $2x\dfrac{dy}{dx} + y = 6x^3$	II. $\dfrac{1}{y}$
C. $ydx - xdy + (\log_e x)dx = 0$	III. $e^{2x}$
D. $ydx - (x+2y^2)dy = 0$	IV. $\sqrt{x}$

Choose the correct answer from the options given below:

2026: 29 May Shift 2

The order of the differential equation $\left(\frac{d^3y}{dx^3}\right)^4 + \frac{d^4y}{dx^4} + \log\left(\frac{dy}{dx}\right) = 0$ is

2022: 25 May Shift 1

The differential equation $\frac{dy}{dx} + \frac{x}{y} = 0$ , represents the family of curves:

2025: 16 May Shift 1

The degree of the differential equation $\left(2 + \left(\frac{dy}{dx}\right)^2\right)^{\frac{3}{2}} = a^2 \frac{d^2y}{dx^2}$ is:

2025: 14 May Shift 2

Particular solution of the differential equation $x(1 + y^2)dx - y(1 + x^2)dy = 0$ , given $y = 0$ when $x = 1$ , is

2026: 30 May Shift 1

Match the LIST-I with LIST-II

	LIST-I Differential equation		LIST-II Sum of the order and the degree of the differential equation
A.	$\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right) = 1$	I.	$2$
B.	$4 + \left(\dfrac{dy}{dx}\right)^4 = 7\left(\dfrac{d^2y}{dx^2}\right)^3$	II.	$3$
C.	$x^3\left(\dfrac{d^2y}{dx^2}\right)^2 + x\left(\dfrac{dy}{dx}\right)^4 + 1 = 0$	III.	$5$
D.	$\dfrac{dy}{dx} = x^4 e^{-3y}$	IV.	$4$

Choose the correct answer from the options given below: