Calculus

Differential Equations

Medium

core

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

$y=Ce^{\frac{x^2}{2}}$

$y=Ce^{-\frac{x^2}{2}}$

$y=(C-x)e^{\frac{x^2}{2}}$

$y=(x+C)e^{\frac{x^2}{2}}$

✅ Correct Option: 4

Rewrite as $\frac{dy}{dx}-xy=e^{x^2/2}$ , a linear equation with integrating factor $e^{-x^2/2}$ .

Then $\frac{d}{dx}\left(ye^{-x^2/2}\right)=1$ , so $ye^{-x^2/2}=x+C$ and $y=(x+C)e^{x^2/2}$ .

2025: 27 May Shift 1

Match List-I with List-II

List-I	List-II
(A) The degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^2 = x\sin\left(\frac{dy}{dx}\right)$	(I) 4
(B) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{1/4} + x^{1/5} = 0$	(II) 1
(C) The degree of differential equation $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + 6y^5 = 0$	(III) Not defined
(D) The degree of differential equation $1 + \left(\frac{dy}{dx}\right)^4 = 7\left(\frac{d^2y}{dx^2}\right)^3$	(IV) 3

Choose the correct answer from the options given below:

2026: 21 May Shift 1

For the differential equation $\frac{\left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} = k, k \neq 0$ , which one of the following is TRUE ?

2026: 29 May Shift 2

The solution of the differential equation $\frac{dy}{dx} + y = 1, y(1) = 0$ is

2022: 4 Aug Shift 1

Solution of the differential equation $\frac{dy}{dx} = x + xy - (1 + y)$ is:

2026: 22 May Shift 1

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

2025: 30 May Shift 1

The differential equation representing the family of curves $y = Ax + \frac{B}{x}$ , $x \neq 0$ where A and B are arbitrary constants, is given by

General solution of the differential equation dydx=ex22+xy\frac{dy}{dx}=e^{\frac{x^2}{2}}+xydxdy​=e2x2​+xy is: (where C is an arbitrary constant)

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CUET Mathematics Past Year Questions (Topic-Wise)

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