The Wronskian & Variation of Parameters · Superchallenge problem

Consider $t^2y''+ty'+\left(t^2-\dfrac14\right)y=t^{3/2}$ on $t>0$ (Bessel's operator of order $\tfrac12$, forced). Let $y$ be the solution with $\displaystyle\lim_{t\to0^+}y(t)=0$ and $y\!\left(\dfrac{\pi}{2}\right)=0$. Find $y(\pi)$.