Monotonic property of solution of a linear (second-order) differential equation Consider the following differential equation
\begin{equation}
y''+q(t) y=0 ,
\end{equation}
where q(t) is a continuous function $\leq 0\;\; \forall t \in \mathbb{R}$. 
I'm trying to prove that each non-constant solution such that $y(0)=0$ is strictly monotone. 
I have tried integrating between 0 and t but in this way I can't show anything useful in fact by using the weighted average theorem I get
\begin{equation}
y'(t)=y'(0)-y(\xi_t)\int_{0}^{t}q(x)dx .
\end{equation}
I'm wondering if I should continue to prove that $y'>$ or $<0$ in a dense subset of $\mathbb{R}$ or if I should use a different approach.

Hint: Show that if there is $a\in {\Bbb R}$ for which $y(a)>0$ (or $y(a)<0)$ and $y'(a)=0$ then $(a,y(a))$ must be a global minimum (or maximum) and get a contradiction.
