The Wronskian & Variation of Parameters · Superchallenge problem

Let $q$ be continuous on $[0,\infty)$ and let $y$ solve $y''+q(t)y=0$ with $y(0)=1$, $y'(0)=0$. Suppose $y(t)>0$ for all $t\ge0$ and $\displaystyle\int_0^\infty\dfrac{dt}{y(t)^2}=3$. There is exactly one solution $u$ of the same equation with $u(0)=1$ and $\displaystyle\lim_{t\to\infty}\dfrac{u(t)}{y(t)}=0$. Find $u'(0)$.