The particular solution of the differential equation dy=e^(2x+y)dx, y(0)=0 is Option 1: x=_e (3+e^2y)/2 Option 2: x=_e (3-e^2y)/2 Option 3: y=_e 2/(3-e^2x) Option 4: y=_e 2/(3+e^2x)

Answer: y=_e 2/(3-e^2x). Solution: Separating gives e^(-y)dy=e^2xdx, so -e^(-y)=e^2x/2+C. Using y(0)=0 gives C=-3/2, hence e^(-y)=(3-e^2x)/2 and y=_e, 2/(3-e^2x), .

Calculus

Differential Equations

Medium

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

$x=\log_e\left|\frac{3+e^{2y}}{2}\right|$

$x=\log_e\left|\frac{3-e^{2y}}{2}\right|$

$y=\log_e\left|\frac{2}{3-e^{2x}}\right|$

$y=\log_e\left|\frac{2}{3+e^{2x}}\right|$

✅ Correct Option: 3

Separating gives $e^{-y}dy=e^{2x}dx$ , so $-e^{-y}=\frac{e^{2x}}{2}+C$ .

Using $y(0)=0$ gives $C=-\frac{3}{2}$ , hence $e^{-y}=\frac{3-e^{2x}}{2}$ and $y=\log_e\left|\frac{2}{3-e^{2x}}\right|$ .

2022: 4 Aug Shift 1

The solution of the differential equation $\frac{dy}{dx} = \frac{\lambda^2}{(x+y)^2}$ ( $\lambda$ is constant) is:

2026: 29 May Shift 2

Match the LIST-I with LIST-II

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

Choose the correct answer from the options given below:

2026: 25 May Shift 1

If m and n are the order and the degree of the differential equation :

$x\dfrac{d^2y}{dx^2} = \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{1/3}$ then which one of the following is TRUE?

2025: 13 May Shift 1

The particular solution of the differential equation $e^x\sqrt{1-y^2}dx + \frac{y}{x}dy = 0$ , given that $y = 1$ , when $x = 0$ is:

2026: 25 May Shift 1

Match the LIST-I with LIST-II

LIST-I Differential equation	LIST-II Integrating factor of the differential equation
A. $\dfrac{dy}{dx} + 2y = xe^{4x}$	I. $\dfrac{1}{x}$
B. $2x\dfrac{dy}{dx} + y = 6x^3$	II. $\dfrac{1}{y}$
C. $ydx - xdy + (\log_e x)dx = 0$	III. $e^{2x}$
D. $ydx - (x+2y^2)dy = 0$	IV. $\sqrt{x}$

Choose the correct answer from the options given below:

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 particular solution of the differential equation dy=e2x+ydx,y(0)=0dy=e^{2x+y}dx, y(0)=0dy=e2x+ydx,y(0)=0 is

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CUET Mathematics Past Year Questions (Topic-Wise)

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