Q.The number of arbitrary constants in the particular solution of a differential equation of third order are: (A) 3 (B) 2 (C) 1 (D) 0

The key idea is that a particular solution contains no arbitrary constants — it is obtained from the general solution by assigning specific values to all constants using given initial or boundary conditions.

• A third-order differential equation has a general solution with 3 arbitrary constants.
• A particular solution is one where all these constants have been replaced by fixed numbers. …

Why the answer is zero — the core idea

The confusion here usually comes from mixing up the general solution with the particular solution. A third-order differential equation has a general solution containing three arbitrary constants (because you need to integrate three times). But the question asks about the particular solution — that’s a completely different beast.

A particular solution is obtained by plugging specific initial or boundary conditions into the general solution, which fixes every constant to a definite number. Once those constants become numbers, they are no longer “arbitrary.” So the count drops to zero.

Let’s walk through it step by step.

1. General solution of a third-order DE A third-order differential equation involves the third derivative. Solving it requires three integrations, each introducing one arbitrary constant. So the general solution looks like:

where $C_1, C_2, C_3$ are arbitrary constants. That’s three constants — but this is not the particular solution.

2. What makes a solution “particular”?

   To get a particular solution, you need extra information — typically three initial conditions (like $y(x_0)=a$, $y'(x_0)=b$, $y''(x_0)=c$) or boundary conditions. You substitute these into the general solution (and its derivatives) to solve for $C_1, C_2, C_3$.

3. Constants become fixed numbers

   After applying the conditions, each constant takes a specific numerical value. For example, you might get $C_1 = 5$, $C_2 = -2$, $C_3 = 0$. These are no longer “arbitrary” — they are determined uniquely by the problem. …

Method: General solution vs particular solution — counting constants

Use this reasoning whenever a question contrasts the general and particular solution and asks how many arbitrary constants remain.

Steps

Step 1: Recall what "general" means

The general solution of an order-$n$ equation contains $n$ arbitrary constants $C_1,\dots,C_n$ — one per integration.

Step 2: Recall what "particular" means

A particular solution is obtained by imposing initial/boundary conditions that pin every constant to a definite number. Once a constant equals a specific value, it is no longer arbitrary.

Step 3: Count what is left …

Common Mistakes

Mistake 1: Answering with the order (3) instead of 0

Why it's wrong: the question asks about the particular solution, where initial/boundary conditions have already fixed every constant to a number. Correct approach: recognise "particular" means no arbitrary constants remain, so the answer is $0$.

Mistake 2: Believing a particular solution keeps at least one free constant …