Calculus

Differential Equations

Easy

core

The number of arbitrary constants in the general solution of a differential equation of order 4 and degree 1 is

✅ Correct Option: 2

For a differential equation:

Order = The highest derivative in the equation

Degree = The power of the highest order derivative (after removing fractions/radicals)

In this problem:

Order = 4 (contains fourth derivative $\frac{d^4y}{dx^4}$ )
Degree = 1 (fourth derivative appears to power 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 constants.

For this differential equation:

Order = 4

Therefore, number of arbitrary constants = 4

When solving a differential equation of order 4, integration is performed 4 times:

First integration introduces constant $C_1$
Second integration introduces constant $C_2$
Third integration introduces constant $C_3$
Fourth integration introduces constant $C_4$

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

Number of arbitrary constants = 4

2025: 30 May Shift 2

The solution of the differential equation $\frac{dy}{dx} = \sqrt\frac{{y}}{x}$ is

2025: 14 May Shift 1

Consider the differential equation, $x\frac{dy}{dx} = y(\log_e y - \log_e x + 1)$ , then which of the following are true?

(A) It is a linear differential equation

(B) It is a homogenous differential equation

(D) Its general solution is $\log_e\left(\frac{x}{y}\right) = Cy$ , where C is constant of integration

(E) If $y(1) = 1$ , then its particular solution is $y = x$

Choose the correct answer from the options given below:

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:

2025: 3 June Shift 2

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

2026: 25 May Shift 1

General solution of the differential equation $(e^x + e^{-x})dy = (3e^{2x} + 3e^{4x})dx$ is:

[Where C is an arbitrary constant]

2025: 15 May Shift 2

Consider the curve which is represented by the differential equation $\frac{dy}{dx} = 1 + x + y + xy$ . If it passes through the point $(0,0)$ , then which of the following is/are true?

(A) it is a straight line.

(B) it is a parabola.

(D) Its equation is $xy(x + 1)\left(y - \frac{1}{\sqrt{e}} + 1\right) = 0$

Choose the correct answer from the options given below:

The number of arbitrary constants in the general solution of a differential equation of order 4 and degree 1 is

Solution

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