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:

• A.
• B.
• C.
• D.

Hint:

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

Answer 1

The Correct Option is D

The differential equation is: \[ \sqrt{4 - x^2} \frac{dy}{dx} = \left( \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - y \right) \sin^{-1} \left( \frac{x}{2} \right) \] Rearranging the terms, we integrate to solve for \( y(x) \): \[ y = \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - 2 + c \cdot e \] Given that \( y(2) = \frac{\pi^2}{4} - 2 \), we solve for \( c \): \[ y(2) = \frac{\pi^2}{4} - 2 \implies c = 0 \] Thus, \( y(0) = -2 \).