The Wronskian & Variation of Parameters · Superchallenge problem

Let $y$ solve $y''-2ty'-2y=0$ with $y(0)=0$, $y'(0)=1$. Given that $y_1=e^{t^2}$ is one solution of the equation, find $\displaystyle\lim_{t\to\infty}e^{-t^2}y(t)$.