Laplace II: Steps, Impulses & Convolution · Hard problem

An undamped spring at rest, $y''+y=\delta\!\left(t-\dfrac{\pi}{2}\right)$, $y(0)=y'(0)=0$, is struck once by a unit hammer blow at $t=\dfrac{\pi}{2}$. Find its velocity $y'\!\left(\dfrac{2\pi}{3}\right)$.