Calculus

Differential Equations

Easy

core

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

✅ Correct Option: 1

A differential equation of order $n$ has a general solution containing $n$ arbitrary constants.

Given differential equation has order $= 4$

The general solution contains 4 arbitrary constants $(C_1, C_2, C_3, C_4)$

A particular solution is obtained by assigning specific values to all arbitrary constants in the general solution.

For example:

General solution: $y = C_1e^x + C_2e^{-x} + C_3\sin(x) + C_4\cos(x)$

Particular solution: $y = 2e^x + 3e^{-x} + \sin(x) + 5\cos(x)$

After assigning specific values to all constants, no arbitrary constants remain in the particular solution.

Number of arbitrary constants in particular solution $= 0$

The degree of the differential equation (which is 3 in this problem) does not affect the number of arbitrary constants.

Therefore, the number of arbitrary constants in the particular solution is $0$ .

2025: 22 May Shift 1

Match List-I with List-II

List-I	List-II
Differential Equation	Degree
(A) $xy\frac{d^2y}{dx^2} + x\left(\frac{dy}{dx}\right)^2 - y\frac{dy}{dx} = 0$	(I) 3
(B) $\frac{d^2y}{dx^2} + \log\left(\frac{dy}{dx}\right) = 0$	(II) 1
(C) $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + \frac{dy}{dx} + 1 = 0$	(III) not defined
(D) $2x^2\left(\frac{d^2y}{dx^2}\right)^3 - 5\left(\frac{dy}{dx}\right)^3 + y = 0$	(IV) 2

Choose the correct answer from the options given below:

2025: 22 May Shift 2

The solution of the differential equation $\frac{dy}{dx} = (1 + x^2)(1 + y^2)$ is (Here C is an arbitrary constant)

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 $\log x=\log_e x$ )

2026: 14 May Shift 2

The solution of the differential equation $\frac{dy}{dx} = x^2 + x + \frac{1}{x}$ is

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: 3 June Shift 2

The general solution of the differential equation $\frac{dy}{dx} = -4xy^2$ is given by

Solution