Calculus

Differential Equations

Easy

core

The degree of the differential equation $\left[1 + \left(\frac{dy}{dx}\right)\right]^3 = \left(\frac{d^2 y}{dx^2}\right)^2$ is :

✅ Correct Option: 2

The given equation is already polynomial in derivatives. Highest order is $\frac{d^2 y}{dx^2}$ and its highest power is 2, so degree $= 2$ .

2026: 30 May Shift 1

The particular solution of the differential equation $\frac{dy}{dx}=x(2\log x+1)$ , given that $y=0$ when $x=2$ , is:

(consider $\log x=\log_e x$ )

2025: 27 May Shift 1

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

2026: 14 May Shift 2

Match the LIST-I with LIST-II

	LIST-I (Differential Equation)		LIST-II (Order and degree)
A.	$\log\left(\dfrac{dy}{dx}\right) + \dfrac{d^2y}{dx^2} = 0$	I.	Order : 3 and degree : not defined
B.	$\dfrac{d^3y}{dx^3} + y^2 + e^{\frac{dy}{dx}} = 4$	II.	Order : 2 and degree : 2
C.	$\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)^2 + \dfrac{dy}{dx} = \sqrt{x}$	III.	Order : 2 and degree : not defined
D.	$\left(1+\left(\dfrac{dy}{dx}\right)^2\right)^{3/2} = \dfrac{d^2y}{dx^2}$	IV.	Order : 2 and degree : 1

Choose the correct answer from the options given below:

2025: 3 June Shift 1

Consider the differential equation $xdy = (y + 2x^3)dx$ . Then which of the following are TRUE?

(A) It is a homogeneous differential equation.

(B) Product of the order and degree of the differential equation in one.

(D) General solution of the differential equation is $y = x^3 + Cx$ , where C is an arbitary constant.

Choose the correct answer from the options given below:

2025: 29 May Shift 2

The particular solution of the differential equation $\frac{dy}{dx} = 8yx$ when $y = 1$ at $x = 0$

2025: 14 May Shift 1

The integrating factor of the differential equation $(x log_e x)\frac{dy}{dx} + y = 2log_e x$ is

The degree of the differential equation [1+(dydx)]3=(d2ydx2)2\left[1 + \left(\frac{dy}{dx}\right)\right]^3 = \left(\frac{d^2 y}{dx^2}\right)^2[1+(dxdy​)]3=(dx2d2y​)2 is :