Question

The number of arbitrary constants in the general solution and in the particular solution of a differential equation of fourth order are respectively

• A. \(0,4\)
• B. \(4,4\)
• C. \(4,0\)
• D. \(0,0\)

Hint:

For a differential equation of order \(n\): \[ \text{General Solution} \rightarrow n \text{ arbitrary constants} \] \[ \text{Particular Solution} \rightarrow 0 \text{ arbitrary constants} \] Example: \[ \frac{d^2y}{dx^2}=0 \] General solution: \[ y=C_1x+C_2 \] contains \(2\) arbitrary constants.

Answer 1

The Correct Option is C

Step 1: Recall what order means.
The order of a differential equation is the highest derivative that appears in it. Here the order is $4$.

Step 2: Link order to arbitrary constants.
The general solution of an order-$n$ differential equation carries exactly $n$ free (arbitrary) constants, one for each time we would integrate.

Step 3: Count the constants in the general solution.
Since the order is $4$, the general solution has $4$ arbitrary constants.

Step 4: Understand the particular solution.
A particular solution comes from the general one after we fix every constant using given conditions.

Step 5: Count the constants in the particular solution.
Once all constants are pinned to specific numbers, none stay arbitrary. So the particular solution has $0$ arbitrary constants.

Step 6: State the pair.
Therefore the answer, general then particular, is $4$ and $0$. \[ \boxed{4,\ 0} \]