Question

Differential equation of family \(y=a\cos \mu x + b\sin \mu x\) is

• A. \(\frac{d^2y}{dx^2} + \mu y = 0\)
• B. \(\frac{d^2y}{dx^2} - \mu^2 y = 0\)
• C. \(\frac{d^2y}{dx^2} + \mu^2 y = 0\)
• D. None of these

Hint:

Any combination of \(\sin\) and \(\cos\) leads to \(y'' + k^2 y = 0\).

Answer 1

The Correct Option is C

Concept: Eliminate constants \(a\) and \(b\) by differentiating twice.

Step 1: Given function. \[ y = a\cos \mu x + b\sin \mu x \]

Step 2: First derivative. \[ \frac{dy}{dx} = -a\mu \sin \mu x + b\mu \cos \mu x \]

Step 3: Second derivative. \[ \frac{d^2y}{dx^2} = -a\mu^2 \cos \mu x - b\mu^2 \sin \mu x = -\mu^2 (a\cos \mu x + b\sin \mu x) \]

Step 4: Substitute. \[ \frac{d^2y}{dx^2} = -\mu^2 y \Rightarrow \frac{d^2y}{dx^2} + \mu^2 y = 0 \]