Question

If \( y = 5\cos x - 3\sin x \), then \( \frac{d^2y}{dx^2} + y \) equals:

• A. \( 8\sin x \cos x \)
• B. \( 3\sin x \cos x \)
• C. \( 1\)
• D. \(0 \)

Hint:

Verify solutions to differential equations by substituting derivatives back into the equation.

Answer 1

The Correct Option is D

We are given: \[ \frac{dy}{dx} = -5\sin x - 3\cos x. \] Step 1: Compute the second derivative Differentiating \(\frac{dy}{dx}\) with respect to \(x\), we get: \[ \frac{d^2y}{dx^2} = -5\cos x + 3\sin x. \]
Step 2: Add \(y\) to the second derivative We know \(y = 5\cos x - 3\sin x\). Adding \(y\) to \(\frac{d^2y}{dx^2}\), we have: \[ \frac{d^2y}{dx^2} + y = (-5\cos x + 3\sin x) + (5\cos x - 3\sin x). \]
Step 3: Simplify the expression Combine like terms: \[ \frac{d^2y}{dx^2} + y = -5\cos x + 5\cos x + 3\sin x - 3\sin x = 0. \]
Conclusion: The equation is satisfied, confirming that: \[ \frac{d^2y}{dx^2} + y = 0. \]