Question

If \( y = y(x) \) is the solution of the differential equation, \[ \sqrt{4 - x^2} \frac{dy}{dx} = \left( \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - y \right) \sin^{-1} \left( \frac{x}{2} \right), \] where \( -2 \leq x \leq 2 \), and \( y(2) = \frac{\pi^2 - 8}{4} \), then \( y^2(0) \) is equal to:

Hint:

When solving differential equations, always ensure proper integration and boundary conditions to find constants of integration.

Answer 1

Correct Answer: 2

Step 1: Given Differential Equation

The differential equation is: \[ \sqrt{4 - x^2} \frac{dy}{dx} = \left( \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - y \right) \sin^{-1} \left( \frac{x}{2} \right) \]

Step 2: Rearrange and Integrate

Rearranging the terms, we integrate to solve for \( y(x) \): \[ y = \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - 2 + c \cdot e \]

Step 3: Solve for \( c \) Using the Initial Condition

Given that \( y(2) = \frac{\pi^2}{4} - 2 \), we solve for \( c \): \[ y(2) = \frac{\pi^2}{4} - 2 \implies c = 0 \]

Step 4: Find \( y(0) \)

Thus, the value of \( y(0) \) is: \[ y(0) = -2 \]

Final Answer: \( y(0) = -2 \)