The general solution of the differential equation dy/dx = e^(x-y) + x^2e^(-y) is equal to: Option 1: e^y = e^x + x^3/3 + c, where c is an arbitrary constant. Option 2: e^y = xe^x + x^2/2 + c, where c is an arbitrary constant. Option 3: e^y = 2e^x + x^2/4 + c, where c is an arbitrary constant. Option 4: y = e^x + x^3/3 + c, where c is an arbitrary constant.

Answer: e^y = e^x + x^3/3 + c, where c is an arbitrary constant. Solution: Given equation: dy/dx = e^(x-y) + x^2e^(-y) Using exponent rules, e^(x-y) = e^x * e^(-y): dy/dx = e^x * e^(-y) + x^2 * e^(-y) Factoring out e^(-y): dy/dx = e^(-y)(e^x + x^2) Separating variables: e^y dy = (e^x + x^2) dx Integrating both sides: e^y dy = (e^x + x^2) dx e^y = e^x + x^3/3 + c where c is the constant of integration. Therefore, the general solution is e^y = e^x + x^3/3 + c

Calculus

Differential Equations

Medium

applied

The general solution of the differential equation $\frac{dy}{dx} = e^{x-y} + x^2e^{-y}$ is equal to:

$e^y = e^x + \frac{x^3}{3} + c$ , where $c$ is an arbitrary constant.

$e^y = xe^x + \frac{x^2}{2} + c$ , where $c$ is an arbitrary constant.

$e^y = 2e^x + \frac{x^2}{4} + c$ , where $c$ is an arbitrary constant.

$y = e^x + \frac{x^3}{3} + c$ , where $c$ is an arbitrary constant.

✅ Correct Option: 1

Given equation: $\frac{dy}{dx} = e^{x-y} + x^2e^{-y}$

Using exponent rules, $e^{x-y} = e^x \cdot e^{-y}$ :

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

Factoring out $e^{-y}$ :

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

Separating variables:

$e^y \, dy = (e^x + x^2) \, dx$

Integrating both sides:

$\int e^y \, dy = \int (e^x + x^2) \, dx$

$e^y = e^x + \frac{x^3}{3} + c$

where $c$ is the constant of integration.

Therefore, the general solution is $e^y = e^x + \frac{x^3}{3} + c$

2026: 26 May Shift 2

Match the LIST-I with LIST-II

	LIST-I Differential equation		LIST-II Order and Degree
A.	$\left(\dfrac{dy}{dx}\right)^2 + 3y\left(\dfrac{d^2y}{dx^2}\right) = 0$	I.	$1,1$
B.	$\sqrt{1-\left(\dfrac{dy}{dx}\right)^2} = a\left(\dfrac{d^2y}{dx^2}\right)^{1/3}$	II.	$1,2$
C.	$y = \dfrac{dy}{dx} + \dfrac{5}{(dy/dx)}$	III.	$2,1$
D.	$\sqrt{1-y^2}\,dx + \sqrt{1-x^2}\,dy = 0$	IV.	$2,2$

Choose the correct answer from the options given below:

2022: 10 Aug Shift 1

If the order and degree of the differential equation $\sqrt{\frac{d^2y}{dx^2}} = \left(1 + \frac{dy}{dx}\right)^{\frac{1}{3}}$ are $a$ and $b$ respectively, then the value of $a^2 + b^2$ is

2026: 19 May Shift 2

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

[where c is an arbitrary constant]

2026: 29 May Shift 1

Which of the following are correct?

(A) The order of the differential equation $\left(\dfrac{dy}{dx}\right)^2+3y^2-5y=0$ is 2.

(B) The degree of the differential equation $x\dfrac{d^2y}{dx^2}=\left(3+2\left(\dfrac{dy}{dx}\right)^2\right)^3$ is 3.

(D) The general solution of an ordinary differential equation of $n^{th}$ order contains $n$ independent arbitrary constants.

Choose the correct answer from the options given below:

2026: 26 May Shift 2

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

2026: 22 May Shift 2

Match List-I with List-II

List-I: Linear differential equation	List-II: Integrating factor
(A) $x\frac{dy}{dx}-y=2x^3$	(I) $e^{-x}$
(B) $2x\frac{dy}{dx}+y=3x^2$	(II) $e^x$
(C) $\frac{dy}{dx}+y=\cos x-\sin x$	(III) $x^{1/2}$
(D) $\frac{dy}{dx}-y=e^x$	(IV) $1/x$

Choose the correct answer from the options given below:

The general solution of the differential equation dydx=ex−y+x2e−y\frac{dy}{dx} = e^{x-y} + x^2e^{-y}dxdy​=ex−y+x2e−y is equal to:

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The general solution of the differential equation $\frac{dy}{dx} = e^{x-y} + x^2e^{-y}$ is equal to: