Question

The differential equation satisfied by \(y = X\sin(6t+5) + Y\cos(6t+5)\) (where \(X\) and \(Y\) are constants) is

• A. \(\frac{d^2y}{dt^2}+6y=0\)
• B. \(\frac{d^2y}{dt^2}=0\)
• C. \(\frac{d^2y}{dt^2}+36y=0\)
• D. \(\frac{d^2y}{dt^2}+25y=0\)

Hint:

Differentiate twice; the second derivative is a constant multiple of y.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The function is a combination of sine and cosine with angular frequency 6, like simple harmonic motion. Differentiating twice brings back the same combination.

Step 2: First derivative
\(\dfrac{dy}{dt}=6X\cos(6t+5)-6Y\sin(6t+5)\).

Step 3: Second derivative
\(\dfrac{d^{2}y}{dt^{2}}=-36X\sin(6t+5)-36Y\cos(6t+5)=-36y\).

Step 4: Equation
\(\dfrac{d^{2}y}{dt^{2}}+36y=0\).

Final Answer:
The differential equation is d^2y/dt^2 + 36y = 0. \[ \boxed{\text{(C) }\dfrac{d^{2}y}{dt^{2}}+36y=0} \]