The Wronskian & Variation of Parameters · Superchallenge problem

Let $q$ be continuous with period $1$, and let $y_1,y_2$ solve $y''+q(t)\,y=0$ with $y_1(0)=1$, $y_1'(0)=0$, $y_2(0)=0$, $y_2'(0)=1$. Suppose $y_1(1)=2$ and $y_2'(1)=3$. There is a nonzero solution $y$ and a number $\lambda>1$ with $y(t+1)=\lambda\,y(t)$ for all $t$. Find $\lambda$.