General solution of the differential equation dy/dx = e^(x^2/2) + xy is Option 1: y = c e^(x^2/2), where c is constant of integration. Option 2: y = c e^(-x^2/2), where c is constant of integration. Option 3: y = (x + c) e^(x^2/2), where c is constant of integration. Option 4: y = (x + c) e^((-x^2)/2), where c is constant of integration.

Answer: y = (x + c) e^(x^2/2), where c is constant of integration. Solution: Rearranging the equation: dy/dx - xy = e^(x^2/2) This is a first-order linear differential equation of the form dy/dx + P(x) y = Q(x), where P(x) = -x and Q(x) = e^(x^2/2). The integrating factor is: I.F. = e^( P(x) dx) = e^( (-x) dx) = e^(-x^2/2) Using the standard solution formula: y * I.F. = Q(x) * I.F. dx + C y * e^(-x^2/2) = e^(x^2/2) * e^(-x^2/2) dx + C Since e^(x^2/2) * e^(-x^2/2) = e^0 = 1, this simplifies to: y * e^(-x^2/2) = 1 dx + C y * e^(-x^2/2) = x + C Multiplying both sides by e^(x^2/2): y = (x + C) e^(x^2/2) where C is an arbitrary constant.

Calculus

Differential Equations

Medium

core

General solution of the differential equation $\frac{dy}{dx} = e^{\frac{x^2}{2}} + xy$ is

$y = c e^{\frac{x^2}{2}}$ , where $c$ is constant of integration.

$y = c e^{-\frac{x^2}{2}}$ , where $c$ is constant of integration.

$y = (x + c) e^{\frac{x^2}{2}}$ , where $c$ is constant of integration.

$y = (x + c) e^{\frac{-x^2}{2}}$ , where $c$ is constant of integration.

✅ Correct Option: 3

Rearranging the equation:

$\frac{dy}{dx} - xy = e^{\frac{x^2}{2}}$

This is a first-order linear differential equation of the form $\frac{dy}{dx} + P(x)\,y = Q(x)$ , where $P(x) = -x$ and $Q(x) = e^{\frac{x^2}{2}}$ .

The integrating factor is:

$\text{I.F.} = e^{\int P(x)\,dx} = e^{\int (-x)\,dx} = e^{-\frac{x^2}{2}}$

Using the standard solution formula:

$y \cdot \text{I.F.} = \int Q(x) \cdot \text{I.F.}\,dx + C$

$y \cdot e^{-\frac{x^2}{2}} = \int e^{\frac{x^2}{2}} \cdot e^{-\frac{x^2}{2}}\,dx + C$

Since $e^{\frac{x^2}{2}} \cdot e^{-\frac{x^2}{2}} = e^{0} = 1$ , this simplifies to:

$y \cdot e^{-\frac{x^2}{2}} = \int 1\,dx + C$

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

Multiplying both sides by $e^{\frac{x^2}{2}}$ :

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

where $C$ is an arbitrary constant.

2026: 29 May Shift 2

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

2022: 16 July Shift 2

The number of arbitrary constants in the general solution of a differential equation of fourth order is:

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:

2022: 25 May Shift 1

The degree of the differential equation

$\left(1 + \frac{dy}{dx}\right)^4 = \left(\frac{d^2y}{dx^2}\right)^2$ is:

2022: 10 Aug Shift 1

General solution of the differential equation $\frac{dy}{dx} + \frac{\sqrt{1 - y^2}}{\sqrt{1 - x^2}} = 0$ is

A. $\tan^{-1} x + \tan^{-1} y = C$

B. $\sin^{-1} x - \cos^{-1} y = C$

C. $x\sqrt{1 - y^2} - y\sqrt{1 - x^2} = C$

D. $\sin^{-1} x + \sin^{-1} y = C$

E. $\cos^{-1} x + \cos^{-1} y = C$

(where C is arbitrary constant)

Choose the correct answer from the options given below

2022: 7 Aug Shift 2

If curve represented by differential equation $x \frac{dy}{dx} + y = e^x$ passes through (1, 1), then $y(-1)$ is :

General solution of the differential equation dydx=ex22+xy\frac{dy}{dx} = e^{\frac{x^2}{2}} + xydxdy​=e2x2​+xy is

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

General solution of the differential equation $\frac{dy}{dx} = e^{\frac{x^2}{2}} + xy$ is