The general solution of the differential equation ydx + (x-y^3)dy = 0 is : Option 1: xy = y^4/4+cx : c is an arbitrary constant. Option 2: x = y^4/4+c : c is an arbitrary constant. Option 3: xy = y^4/4+c : c is an arbitrary constant. Option 4: xy = y^3/3+C : C is an arbitrary constant.

Answer: xy = y^4/4+c : c is an arbitrary constant. Solution: Treat x as the dependent variable: dx/dy+x/y=y^2. The integrating factor is y, so d/dy(xy)=y^3 and integration gives xy=y^4/4+c.

Calculus

Differential Equations

Medium

core

The general solution of the differential equation $ydx + (x-y^3)dy = 0$ is :

$xy = \dfrac{y^4}{4}+cx$ : $c$ is an arbitrary constant.

$x = \dfrac{y^4}{4}+c$ : $c$ is an arbitrary constant.

$xy = \dfrac{y^4}{4}+c$ : $c$ is an arbitrary constant.

$xy = \dfrac{y^3}{3}+C$ : $C$ is an arbitrary constant.

✅ Correct Option: 3

Treat $x$ as the dependent variable: $\dfrac{dx}{dy}+\dfrac{x}{y}=y^2$ . The integrating factor is $y$ , so $\dfrac{d}{dy}(xy)=y^3$ and integration gives $xy=\dfrac{y^4}{4}+c$ .

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:

2025: 27 May Shift 1

Which one of the following equations is a homogeneous differential equation?

2026: 31 May Shift 2

Match the LIST-I with LIST-II

[ $a$ is an arbitrary constant]

	LIST-I Differential Equation		LIST-II Family of curves
A.	$x\,dy - y\,dx = 0$	I.	$x^2 + y^2 = a^2$
B.	$2y\,dx - x\,dy = 0$	II.	$y = ax$
C.	$x\,dx + y\,dy = 0$	III.	$y^2 = 16x + a$
D.	$y\,dy - 8\,dx = 0$	IV.	$x^2 = ay$

Choose the correct answer from the options given below:

2025: 16 May Shift 1

The particular solution of the differential equation $\log\left(\frac{dy}{dx}\right)= 3x + 4y$ satisfying $y = 0$ when $x = 0$ is:

2026: 22 May Shift 2

Let y(x) be the solution curve of the differential equation $x(1+y^2)dx - y(1+x^2)dy = 0$ and it passes through the point (0, 1), then

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 ydx+(x−y3)dy=0ydx + (x-y^3)dy = 0ydx+(x−y3)dy=0 is :

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The general solution of the differential equation $ydx + (x-y^3)dy = 0$ is :