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

Concept: For a differential equation of order \(n\),

• The

general solution contains \(n\) arbitrary constants.

• The

particular solution is obtained after assigning specific values to all arbitrary constants and therefore contains no arbitrary constant.

Step 1: Determine the number of arbitrary constants in the general solution. Given that the differential equation is of fourth order, \[\begin{aligned} n=4 \end{aligned}\] Hence, the general solution contains \[\begin{aligned} 4 \end{aligned}\] arbitrary constants.

Step 2: Determine the number of arbitrary constants in the particular solution. A particular solution is obtained by fixing all arbitrary constants using given conditions. Therefore, \[\begin{aligned} \text{Number of arbitrary constants}=0 \end{aligned}\]

Step 3: Write the required pair. \[ \begin{aligned} \text{General Solution} &:\quad 4 \text{ arbitrary constants} \\ \text{Particular Solution} &:\quad 0 \text{ arbitrary constants} \end{aligned} \] \[\begin{aligned} \boxed{(4,0)} \end{aligned}\] Hence, option \(\mathbf{(C)}\) is correct.