Calculus

Differential Equations

Medium

common

The solution of the differential equation $9+y^2\dfrac{dy}{dx}=x^2$ is :

[Where $C$ is an arbitrary constant]

$9y+\dfrac{y^3}{3}=\dfrac{x^3}{3}+C$

$x^3-y^3=27x+C$

$x^3-y^3=27y+C$

$9y+\dfrac{x^3}{3}=\dfrac{y^3}{3}+C$

✅ Correct Option: 2

Separating the variables gives $y^2\,dy = (x^2-9)\,dx$ .

Integrating, $\dfrac{y^3}{3} = \dfrac{x^3}{3} - 9x + C$ .

Multiplying by 3 and rearranging gives $x^3 - y^3 = 27x + C$ .

Note: The equation is printed without brackets, so we read only $y^2$ as multiplying $\dfrac{dy}{dx}$ , not the whole of $9+y^2$ .

2026: 18 May Shift 1

Match List-I with List-II

List-I Differential equation	List-II Order and degree
(A) $\frac{d^4y}{dx^4} + \sin\left(\frac{d^3y}{dx^3}\right) = 0$	(I) order = 2, degree = 1
(B) $\left(\frac{d^2y}{dx^2}\right)^2 + \frac{dy}{dx} = 0$	(II) order = 3, degree = 1
(C) $\frac{d^3y}{dx^3} + y^2 + \frac{dy}{dx} = 0$	(III) order = 2, degree = 2
(D) $3y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^4 = 0$	(IV) order = 4, degree not defined

Choose the correct answer from the options given below:

2025: 19 May Shift 1

The general solution of the differential equation $\frac{dy}{dx} = e^{ax+by}$ is: (Here C is an arbitrary constant)

2026: 18 May Shift 1

The differential equation satisfied by the equation $y = ae^{bx}$ , where $a$ and $b$ are arbitrary constants, is

2026: 14 May Shift 2

Match the LIST-I with LIST-II

LIST-I		LIST-II
A.	The degree of the differential equation $\left(\frac{d^3y}{dx^3}\right)^4+\left(\frac{d^2y}{dx^2}\right)^5+\left(\frac{dy}{dx}\right)+y=0$	I.	0
B.	The order of the differential equation $\left(\frac{d^3y}{dx^3}\right)^4+\left(\frac{d^2y}{dx^2}\right)^5+\left(\frac{dy}{dx}\right)+y^2=0$	II.	4
C.	The number of arbitrary constants in the general solution of a differential equation of second order is	III.	3
D.	The number of arbitrary constants in a particular solution of a differential equation of third order is	IV.	2

Choose the correct answer from the options given below:

2025: 26 May Shift 2

The particular solution of the differential equation $\frac{dy}{dx} = e^{x^2/2} + xy$ , when $x = 0$ , $y = 1$ , is

2022: 16 July Shift 2

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

The solution of the differential equation 9+y2dydx=x29+y^2\dfrac{dy}{dx}=x^29+y2dxdy​=x2 is : [Where CCC is an arbitrary constant]