The solution of the differential equation dy/dx = (1 + x^2)(1 + y^2) is (Here C is an arbitrary constant) Option 1: ^(-1) y + x + x^3/3 = C Option 2: ^(-1) y - x - x^3/3 = C Option 3: ^(-1) x - y - y^3/3 = C Option 4: ^(-1) x + y + y^3/3 = C

Answer: ^(-1) y - x - x^3/3 = C. Solution: The differential equation dy/dx = (1 + x^2)(1 + y^2) is a separable differential equation. The right side is a product of a function of x and a function of y. Separating the variables: dy/(1 + y^2) = (1 + x^2)dx Integrating both sides: dy/(1 + y^2) = (1 + x^2)dx For the left side: dy/(1 + y^2) = ^(-1)y For the right side: (1 + x^2)dx = 1 * dx + x^2 dx = x + x^3/3 Combining both sides: ^(-1)y = x + x^3/3 + C Rearranging: ^(-1)y - x - x^3/3 = C Therefore, the solution is ^(-1) y - x - x^3/3 = C

Calculus

Differential Equations

Medium

common

The solution of the differential equation $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$ is (Here C is an arbitrary constant)

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

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

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

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

✅ Correct Option: 2

The differential equation $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$ is a separable differential equation.

The right side is a product of a function of $x$ and a function of $y$ .

Separating the variables:

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

Integrating both sides:

$\int \frac{dy}{(1 + y^2)} = \int (1 + x^2)dx$

For the left side:

$\int \frac{dy}{(1 + y^2)} = \tan^{-1}y$

For the right side:

$\int (1 + x^2)dx = \int 1 \cdot dx + \int x^2 dx$

$= x + \frac{x^3}{3}$

Combining both sides:

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

Rearranging:

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

Therefore, the solution is $\tan^{-1} y - x - \frac{x^3}{3} = C$

2026: 19 May Shift 1

The solution of the initial value problem $x\frac{dy}{dx} = 2, y(1) = 2$ is

2024: 16 May Shift 1

For the differential equation $\left(x \log _{e} x\right) d y=\left(\log _{e} x-y\right) d x$

(A) Degree of the given differential equation is $1$ .

(B) It is a homogeneous differential equation.

(C) Solution is $2y \log _{\mathrm{e}} \mathrm{x}+A=\left(\log _{\mathrm{e}} \mathrm{x}\right)^{2}$ , where $A$ is an arbitrary constant

(D) Solution is $2 y \log _{e} x+A=\log _{e}\left(\log _{e} x\right)$ , where $A$ is an arbitrary constant

Choose the correct answer from the options given below :

2025: 3 June Shift 1

Match List-I with List-II

List-I	List-II
Differential Equation	Sum of order and degree
(A) $\frac{d^2y}{dx^2} + \frac{dy}{dx} + 3y = \sin x$	(I) 2
(B) $\frac{dy}{dx} = \sin(x + y)$	(II) 3
(C) $\sqrt{1 + (\frac{dy}{dx})^2} = \frac{d^2y}{dx^2}$	(III) 4
(D) $x^2(\frac{d^2y}{dx^2})^3 + y(\frac{dy}{dx})^4 + y^5 = 0$	(IV) 5

Choose the correct answer from the options given below:

2026: 12 May Shift 2

Which of the following differential equations has $y=x$ as one of the particular solution?

2026: 31 May Shift 2

Match the LIST-I with LIST-II

	LIST-I Differential equation		LIST-II Integrating factor
A.	$\dfrac{dy}{dx} + 3y = e^{-2x}$	I.	$\dfrac{1}{x}$
B.	$x\dfrac{dy}{dx} + y = 3x^2$	II.	$e^{-x}$
C.	$x\dfrac{dy}{dx} - y = 3x^2$	III.	$x$
D.	$\dfrac{dy}{dx} - y = x$	IV.	$e^{3x}$

Choose the correct answer from the options given below:

2023: 23 May Shift 3

Which of the following differential equation represents the family of circles touching the x-axis at the origin ?

The solution of the differential equation dydx=(1+x2)(1+y2)\frac{dy}{dx} = (1 + x^2)(1 + y^2)dxdy​=(1+x2)(1+y2) is (Here C is an arbitrary constant)

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The solution of the differential equation $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$ is (Here C is an arbitrary constant)