Question

The number of arbitrary constants in a general solution of a differential equation of fifth order is

• A. 0
• B. 4
• C. 3
• D. 5

Hint:

This is a direct-knowledge question. Remember this simple rule: Number of arbitrary constants in the general solution = Order of the differential equation. A "particular solution" is obtained by finding specific values for these constants and has zero arbitrary constants.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The order of a differential equation is the order of the highest derivative that appears in the equation. The general solution to a differential equation is a solution that contains a number of arbitrary constants.
Step 2: Detailed Explanation:
A fundamental theorem in the study of ordinary differential equations states that the general solution of an \(n\)-th order differential equation will contain exactly \(n\) independent arbitrary constants.
The process of solving an \(n\)-th order differential equation involves \(n\) integrations, with each integration introducing one constant of integration.
In this problem, the differential equation is of the fifth order (order = 5).
Therefore, its general solution must contain five arbitrary constants.
Step 3: Final Answer:
The number of arbitrary constants is equal to the order of the differential equation, which is 5. So, the correct option is (iv).