Consider the Hill's Equation $u''+a(t)u=0$ where $a(t)=a(t+T)$ for all $t$. Show that if $a(t)<0$ for all $t$, then the solution satisfying the inital condition $u(0)=u'(0)=1$ is unbounded as $t \rightarrow \infty$. Hence deduce that this solution describes behavior.
So I was given a hint to try to go about this by using the expression $u'=1-\int^t_0a(s)u(s) ds$
I figured, that if I can use this to show that $u'(t) \geq 1$ for $t \geq 0$, then that should be sufficient, but after playing around with the expression above I haven't been able to make any progress, so if at the very least I could get a few steps in the right direction that would be most helpful. Thanks.
Hint: if $t_0$ is the least $t > 0$ such that $u'(t) \le 1/2$, then $u(s) \ge 1 + s/2$ for $0 \le s \le t_0$, so ...