Question

The differential equation of the family of curves $y = a\cos\mu x + b\sin\mu x$, where $a$ and $b$ are arbitrary constants, is

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

Hint:

The family $y = a\cos\mu x + b\sin\mu x$ satisfies the simple harmonic equation $y'' + \mu^2 y = 0$. This is a standard result.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Differentiate twice to eliminate both constants $a$ and $b$.
Step 2: Detailed Explanation:
$y' = -a\mu\sin\mu x + b\mu\cos\mu x$.
$y'' = -a\mu^2\cos\mu x - b\mu^2\sin\mu x = -\mu^2(a\cos\mu x + b\sin\mu x) = -\mu^2 y$.
Hence $y'' + \mu^2 y = 0$.
Step 3: Final Answer:
$\dfrac{d^2y}{dx^2} + \mu^2 y = 0$.