Step 1: Understanding the Concept:
To form a differential equation, we differentiate the equation of the family of curves as many times as the number of arbitrary constants (here, 2: \( a \) and \( b \)) and then eliminate these constants. Step 2: Detailed Explanation:
Given: \( y = a \cos \mu x + b \sin \mu x \).
Differentiating once with respect to \( x \):
\[ \frac{dy}{dx} = -a \mu \sin \mu x + b \mu \cos \mu x \]
Differentiating again with respect to \( x \):
\[ \frac{d^{2}y}{dx^{2}} = -a \mu^{2} \cos \mu x - b \mu^{2} \sin \mu x \]
Factor out \( -\mu^{2} \):
\[ \frac{d^{2}y}{dx^{2}} = -\mu^{2} (a \cos \mu x + b \sin \mu x) \]
Substituting the original expression for \( y \):
\[ \frac{d^{2}y}{dx^{2}} = -\mu^{2} y \]
Rearranging:
\[ \frac{d^{2}y}{dx^{2}} + \mu^{2} y = 0 \]. Step 3: Final Answer:
The differential equation is \( \frac{d^{2}y}{dx^{2}} + \mu^{2} y = 0 \).