The Wronskian & Variation of Parameters · Superchallenge problem

A lightly driven oscillator obeys $y''+y=t^2e^{-t}$ with $y(0)=0$, $y'(0)=0$. As $t\to\infty$ the motion settles into a pure oscillation. Find $\displaystyle\limsup_{t\to\infty}y(t)$, its amplitude.