If \( y = y(x) \) is the solution of the differential equation, \(\displaystyle \sqrt{4 - x^2} \frac{dy}{dx} = \left( \left( \sin^{-1} \left( \frac{x}{2} \right) \right)^2 - y \right) \sin^{-1} \left( \frac{x}{2} \right) \), \( -2 \le x \le 2 \), \( \displaystyle y(2) = \frac{\pi^{2} - 8}{4} \), then \( y^{2}(0) \) is equal to _________.
