The number of arbitrary constants in the general solution of a differential equation of fourth order are:
Explanation
The number of arbitrary constants in the general solution of a differential equation is equal to the order of the differential equation. For a fourth-order differential equation, the general solution will contain exactly 4 arbitrary constants. This is because solving a differential equation of order n requires integrating n times, and each integration introduces one arbitrary constant. The context shows examples of first-order equations (like y = x² + 2x + C and y = cos x + C) having exactly one arbitrary constant in their general solutions, confirming this principle.
Solution Steps
Step 1: Identify that the question asks about the number of arbitrary constants in the general solution of a fourth-order differential equation.
Step 2: Apply the fundamental principle that the number of arbitrary constants in a general solution equals the order of the differential equation.
Step 3: Since the order is 4, the number of arbitrary constants is 4.
Step 4: Select option (D) 4 as the correct answer.