Calculus

Differential Equations

Medium

applied

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?

m = n

m + 2n = 7

2m + n = 8

3m + n = 9

✅ Correct Option: 4

Highest derivative is $\dfrac{d^2y}{dx^2}$ , so $m=2$ .

Cubing both sides removes the fractional power and gives $x^3\left(\dfrac{d^2y}{dx^2}\right)^3$ , so $n=3$ .

Then $3m+n=6+3=9$ .

2026: 18 May Shift 2

The order and degree of the differential equation $\left[2+\left(\dfrac{dy}{dx}\right)\right]^{5/2}=7\dfrac{d^3y}{dx^3}$ is

2026: 21 May Shift 1

The solution of the differential equation $e^{\frac{dy}{dx}} = x+1$ ; $y(0)=5$ is

2025: 21 May Shift 2

Match List-I with List-II

List-I	List-II
(A) Integrating factor of $xdy - (y + x^2)dx = 0$	(I) $x^2$
(B) Integrating factor of $xdy + (2y + x^2)dx = 0$	(II) $x^3$
(C) Integrating factor of $(3y - x^2)dx + xdy = 0$	(III) $x$
(D) Integrating factor of $(y + 3x^2)dx + xdy = 0$	(IV) $\frac{1}{x}$

Choose the correct answer from the options given below:

2026: 29 May Shift 1

The general solution of the differential equation $(x+1)\dfrac{dy}{dx}=2e^{-y}-1$ is:

(where C is an arbitrary constant)

2026: 29 May Shift 2

The order of the differential equation $\left(\frac{d^3y}{dx^3}\right)^4 + \frac{d^4y}{dx^4} + \log\left(\frac{dy}{dx}\right) = 0$ is

2023: 22 May Shift 3

The differential equation $y = xp + \sqrt{x^2 p^3 + 4}$ where $p = \frac{dy}{dx}$ is :

(A) of order 1

(B) of degree 1

(D) of degree 3

Choose the correct answer from the options given below :

If m and n are the order and the degree of the differential equation : xd2ydx2=[1+(dydx)2]1/3x\dfrac{d^2y}{dx^2} = \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{1/3}xdx2d2y​=[1+(dxdy​)2]1/3 then which one of the following is TRUE?