Calculus

Differential Equations

Easy

common

The number of arbitrary constants in the general solution of a differential equation with degree 1 and order 3, is

✅ Correct Option: 3

Given:

Order = 3

Degree = 1

The number of arbitrary constants in the general solution of a differential equation equals the order of the differential equation.

The degree does not affect the number of arbitrary constants.

Number of arbitrary constants = Order = 3

When solving a differential equation of order 3, integration is performed 3 times. Each integration introduces one arbitrary constant.

For example, if $\frac{d^3y}{dx^3} = 0$ :

First integration: $\frac{d^2y}{dx^2} = C_1$

Second integration: $\frac{dy}{dx} = C_1x + C_2$

Third integration: $y = \frac{C_1x^2}{2} + C_2x + C_3$

Therefore, the number of arbitrary constants is 3.

2026: 19 May Shift 1

The solution of the initial value problem $x\frac{dy}{dx} = 2, y(1) = 2$ is

2025: 3 June Shift 2

The number of arbitrary constants in the particular solution of a differential equation of order 4 and degree 3 is

2025: 26 May Shift 2

Match List-I with List-II

List-I	List-II
Differential Equations	Order and degree
(A) $ydx + x\log(y/x)dy - 2xdy = 0$	(I) Order : 2, degree:1
(B) $\left(\frac{d^3y}{dx^3}\right)^2 + 3\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^4 = y^2$	(II) Order :1, degree:1
(C) $\frac{dy}{dx} + \log\left(\frac{dy}{dx}\right) + x = y$	(III) Order : 3, degree:2
(D) $\left(\frac{ds}{dt}\right)^4 + 2s\frac{d^2s}{dt^2} = 0$	(IV) Order : 1, degree: Not defined

Choose the correct answer from the options given below:

2025: 2 June Shift 1

Match List-I with List-II

List-I	List-II
(A) The degree of differential equation $\frac{d^3y}{dx^3} = e^{\frac{dx}{dy}}$	(I) 2
(B) The order of differential equation $\left(\frac{dy}{dx}\right)^2 + \frac{d^3y}{dx^3} = 0$	(II) 4
(C) The sum of order and degree of differential equation $\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right) + \left(\frac{dy}{dx}\right)^5 = x$	(III) not defined
(D) The number of arbitrary constants in the general solution of a differential equation of order 2	(IV) 3

Choose the correct answer from the options given below:

2026: 15 May Shift 1

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

2026: 30 May Shift 2

Match List-I with List-II

List-I	List-II
Curve $y=f(x)$ (where C is an arbitrary constant)	The curve $y=f(x)$ satisfies the differential equation
(A) $y = \log x + C$	(I) $x\frac{dy}{dx} - y = 0$
(B) $y = e^x + C$	(II) $x\frac{dy}{dx} - 1 = 0$
(C) $y = cx$	(III) $x^2\frac{dy}{dx} + 1 = 0$
(D) $y = \frac{1}{x} + C$	(IV) $\frac{d^2y}{dx^2} - \frac{dy}{dx} = 0$

Choose the correct answer from the options given below:

Solution