Let $$y = y(x)$$ be the solution of the differential equation $$x\sqrt{1-x^2} \, dy + \left(y\sqrt{1-x^2} - x\cos^{-1}x\right)dx = 0$$, $$x \in (0,1)$$, $$\lim_{x \to 1^-} y(x) = 1$$. Then $$y\left(\frac{1}{2}\right)$$ equals :

A

$$3 - \frac{\pi}{\sqrt{3}}$$

B

$$4 - \sqrt{3}\pi$$

C

$$4 - \frac{2\pi}{\sqrt{3}}$$

D

$$3 - \frac{\pi}{2\sqrt{3}}$$