Q.(iii) The number of arbitrary constants in the general solution of a differential equation of order three is ______.

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

• The general solution of an $n$th-order differential equation contains exactly $n$ independent arbitrary constants.
• Here, the order is given as three. …

The idea is simple: solving a differential equation means undoing derivatives. Each time you integrate, you introduce one new arbitrary constant. The order of the equation tells you the highest derivative present — so to go from that derivative back to the original function, you need to integrate that many times. That means the general solution will contain exactly that many independent constants.

Let’s walk through it step by step.

1. Order equals the highest derivative.

   A differential equation of order three involves the third derivative, say $\frac{d^3y}{dx^3}$, possibly along with lower derivatives and $y$ itself.

2. Each integration adds one constant.

   To solve, you reverse the differentiation. Starting from the third derivative, the first integration gives the second derivative plus one constant:

3. Integrate again. The second integration gives the first derivative plus a second constant:

4. One more integration. The third integration yields the function $y$ itself, introducing a third constant: $y = \int \left( \frac{dy}{dx} \right) dx + C_3$ …

Method: Relating order to the number of arbitrary constants

Use this recall-and-reason rule linking a DE's order to its general solution.

Steps

Step 1: Recall the defining fact

The general solution of an $n$th-order differential equation contains exactly $n$ independent arbitrary constants — no more, no fewer.

Step 2: Read off the order …

Common Mistakes

Mistake 1: Linking the number of constants to the degree instead of the order

Why it's wrong: the general solution has as many arbitrary constants as the order of the DE, independent o …