Solution of the differential equation (x + xy)dy - y(1 - x^2)dx = 0 is Option 1: y = /y - x^2/2 + C Option 2: y = /y + x^2/2 + C Option 3: y = xy - x^2/2 + C Option 4: y = xy + x^2/2 + C

Answer: y = /y - x^2/2 + C. Solution: (x+xy)dy = y(1-x^2)dx, i.e., x(1+y)dy = y(1-x^2)dx. Separate: (1+y)/ydy = (1-x^2)/xdx = (1/x-x)dx. Integrating: y + y = x - x^2/2 + C, so y = /y - x^2/2 + C.

Calculus

Differential Equations

Medium

Solution of the differential equation $(x + xy)dy - y(1 - x^2)dx = 0$ is

$y = \log\frac{x}{y} - \frac{x^2}{2} + C$

$y = \log\frac{x}{y} + \frac{x^2}{2} + C$

$y = \log xy - \frac{x^2}{2} + C$

$y = \log xy + \frac{x^2}{2} + C$

✅ Correct Option: 1

$(x+xy)dy = y(1-x^2)dx$ , i.e., $x(1+y)dy = y(1-x^2)dx$ .

Separate: $\frac{1+y}{y}dy = \frac{1-x^2}{x}dx = \left(\frac{1}{x}-x\right)dx$ .

Integrating: $\ln y + y = \ln x - \frac{x^2}{2} + C$ , so $y = \ln\frac{x}{y} - \frac{x^2}{2} + C$ .

2025: 19 May Shift 1

Match List-I with List-II

List-I	List-II
Differential Equation	Order and degree of differential equation
(A) $\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^2 = e^{\frac{dy}{dx}} + 1$	(I) Order = 1, Degree = 2
(B) $\left(\frac{d^2y}{dx^2}\right)^2 + 4\left(\frac{dy}{dx}\right)^3 = e^y - 1$	(II) Order = 2, Degree = 1
(C) $3\left(\frac{dy}{dx}\right) + 4y + e^y = \frac{dx}{dy}$	(III) Order = 2, Degree = 2
(D) $\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right) = \left(e^y + \frac{dy}{dx}\right)^2$	(IV) Order = 2, Degree = Not defined

Choose the correct answer from the options given below:

2025: 14 May Shift 1

Which of the following are linear first order differential equations?

(A) $\frac{dy}{dx} + P(x)y = Q(x)$

(B) $\frac{dx}{dy} + P(y)x = Q(y)$

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

Choose the correct answer from the options given below:

2022: 25 May Shift 1

The solution of the differentiable equation $2x\frac{dy}{dx} + y = 14x^3, x > 0$ , is

2023: 22 May Shift 3

Integerating factor of $(x \log_e x) \frac{dy}{dx} + y = 2 \log_e x$ is :

2025: 19 May Shift 1

Match List-I with List-II

List-I	List-II
Differential Equation	Integrating Factor
(A) $y dx + (x - y^3)dy = 0$	(I) $e^{-x}$
(B) $x\frac{dy}{dx} + y = x^2$	(II) $\frac{1}{x}$
(C) $\frac{dy}{dx} - y = e^x$	(III) $y$
(D) $x dy - y dx = x^3 dx$	(IV) $x$

Choose the correct answer from the options given below:

2026: 22 May Shift 1

General solution of the differential equation $\frac{dy}{dx}=e^{\frac{x^2}{2}}+xy$ is: (where C is an arbitrary constant)

Solution of the differential equation (x+xy)dy−y(1−x2)dx=0(x + xy)dy - y(1 - x^2)dx = 0(x+xy)dy−y(1−x2)dx=0 is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

Solution of the differential equation $(x + xy)dy - y(1 - x^2)dx = 0$ is