The Wronskian & Variation of Parameters · Superchallenge problem

An oscillator obeys $y''+y=f(t)$ and starts at rest, $y(0)=y'(0)=0$. During $0\le t\le\dfrac{\pi}{2}$ a control force with $|f(t)|\le1$ may be applied; afterwards $f=0$. What is the largest possible amplitude of the free oscillation for $t>\dfrac{\pi}{2}$?