Q.The number of arbitrary constants in the general solution of a differential equation of fourth order are: (A) 0 (B) 2 (C) 3 (D) 4

The key idea is that the order of a differential equation equals the number of arbitrary constants in its general solution.

• A fourth-order differential equation involves the fourth derivative as the highest derivative.
• The general solution is obtained by integrating four times.
• Each integration introduces one arbitrary constant. …

The key idea here is simple but foundational: the order of a differential equation is the highest derivative present, and that number directly tells you how many independent constants appear in the general solution. Why? Because solving a differential equation essentially means "undoing" derivatives — each integration introduces one new constant. A fourth-order equation requires four integrations to go from the highest derivative back to the original function, so you get four constants.

Let’s walk through it step by step.

1. Understand what "order" means.

   The order of a differential equation is the highest derivative that appears. For example, $y'''' + y''' = 0$ is fourth-order because the highest derivative is $y''''$ (the fourth derivative). This is given in the problem.

2. Link order to integration.

   To find the general solution, you integrate repeatedly. Each integration reduces the derivative by one level and introduces one arbitrary constant. Starting from the fourth derivative:

   • Integrate once: $y'''$ appears, plus constant $C_1$.
   • Integrate again: $y''$ appears, plus constant $C_2$.
   • Integrate again: $y'$ appears, plus constant $C_3$.
   • Integrate a fourth time: $y$ appears, plus constant $C_4$.

3. Count the constants.

   After four integrations, you have four distinct constants: $C_1, C_2, C_3, C_4$. These are arbitrary — they can take any value (subject to initial conditions, if given). The general solution is a family of functions parameterized by these four constants.

4. Why not fewer or more? …

Method: Counting arbitrary constants from the order

Use this reasoning pattern for any "how many arbitrary constants does the general solution have?" question — it is a classification question, not a solving one.

Steps

Step 1: Read off the order

The order of a differential equation is the order of the highest derivative that appears — not the highest power. A fourth-order equation has $\dfrac{d^4y}{dx^4}$ as its top derivative.

Step 2: Link order to integrations

Recovering $y$ from its highest derivative takes one integration per order, and each integration introduces exactly one arbitrary constant. …

Common Mistakes

Mistake 1: Confusing order with degree

Why it's wrong: the number of arbitrary constants is tied to the order (highest derivative), not the degree (its power). A student who looks at a power and answers something other than $4$ has mixed the two up. Correct approach: count only the highest derivative's order — here $4$.

Mistake 2: Thinking constants can be fewer than the order …