Let $y = y(x)$ be the solution of the differential equation <br> $\sin x \frac{dy}{dx} + y \cos x = 4 x, \; x \subset (0, \pi)$. If $y (\frac{\pi}{2})=0$, then $y (\frac{\pi}{6})$ is equal to

Question

( A ) $-\frac{4}{9} \pi^2$
( B ) $-\frac{8}{9} \pi^2$
( C ) $-\frac{8}{9\sqrt{3}} \pi^2$
( D ) $-\frac{4}{9\sqrt{3}} \pi^2$