The Wronskian & Variation of Parameters · Superchallenge problem

The equation $y''-2ty'+2y=0$ has the solution $y_1=t$. Let $y$ be the solution with $y(0)=1$ and $y'(0)=0$. Find $\displaystyle\lim_{t\to\infty}t^2e^{-t^2}\,y(t)$.