Question

The differential equation whose solution is \( y = c_1 \cos ax + c_2 \sin ax \) (where \( c_1 \) and \( c_2 \) are arbitrary constants) is

• A. \( \frac{d^2y}{dx^2} - a^2y = 0 \)
• B. \( \frac{d^2y}{dx^2} + a^2y = 0 \)
• C. \( \frac{d^2y}{dx^2} + y^2 = 0 \)
• D. \( \frac{d^2y}{dx^2} + a^2y = 0 \)

Hint:

For trigonometric solutions, differentiating twice and applying trigonometric identities will lead to a standard form of the differential equation.

Answer 1

The Correct Option is B

Step 1: Differentiate the given solution.
The given solution is \( y = c_1 \cos ax + c_2 \sin ax \). To find the corresponding differential equation, take the first and second derivatives of \( y \) with respect to \( x \).
Step 2: Simplify the expression.
After differentiating twice, you will arrive at the equation \( \frac{d^2y}{dx^2} + a^2y = 0 \), which matches option (B).
Step 3: Conclusion.
The correct answer is (B) \( \frac{d^2y}{dx^2} + a^2y = 0 \).