Question

If \( y = A \cos \theta + B \sin \theta \), then prove that \( \frac{d^2y}{d\theta^2} = -y \).

Hint:

When asked to prove that a function satisfies a differential equation, always find the required derivatives first. Then, substitute them into the equation. Often, you will be able to substitute the original function back into the expressions for the derivatives to simplify the process.

Answer 1

Step 1: Understanding the Concept:
We are given a function y in terms of \( \theta \) and asked to prove it satisfies a second-order differential equation. This requires finding the first and second derivatives of y with respect to \( \theta \) and showing that the second derivative is the negative of the original function.
Step 2: Key Formula or Approach:
1. Find the first derivative, \( \frac{dy}{d\theta} \).
2. Find the second derivative, \( \frac{d^2y}{d\theta^2} \).
3. Show that the expression for \( \frac{d^2y}{d\theta^2} \) is equal to \( -y \).
Step 3: Detailed Explanation or Calculation:
The given function is \( y = A \cos \theta + B \sin \theta \), where A and B are constants.
First Derivative:
Differentiate y with respect to \( \theta \): \[ \frac{dy}{d\theta} = \frac{d}{d\theta}(A \cos \theta + B \sin \theta) = -A \sin \theta + B \cos \theta \] Second Derivative:
Differentiate \( \frac{dy}{d\theta} \) with respect to \( \theta \): \[ \frac{d^2y}{d\theta^2} = \frac{d}{d\theta}(-A \sin \theta + B \cos \theta) = -A \cos \theta - B \sin \theta \] Now, factor out -1 from the expression for the second derivative: \[ \frac{d^2y}{d\theta^2} = -(A \cos \theta + B \sin \theta) \] We can see that the expression in the parenthesis is the original function y. \[ \frac{d^2y}{d\theta^2} = -y \] Step 4: Final Answer:
Hence, it is proved that \( \frac{d^2y}{d\theta^2} = -y \). This is the differential equation for simple harmonic motion.