The Wronskian & Variation of Parameters · Superchallenge problem

Solve $ty''-y'+4t^3y=4t^3\sec(t^2)$ on $0<t<\sqrt{\pi/2}$ for the solution with $\displaystyle\lim_{t\to0^+}y(t)=1$ and $\displaystyle\lim_{t\to0^+}\dfrac{y'(t)}{t}=0$. Find $y\!\left(\sqrt{\pi/3}\right)$.