Prove ode solution has boundary:
Hello, suppose the ode equation as follows $$ \ddot{y} + R(t)y(t) = 0$$ where $R(t)> 0$ bounded, $t\in (t_0,t_2)$.
For discuss convenient, we add some addition condition in $R$, if $t\in (t_0,t_1)$, $\dot{R}\leq 0$, and $t\in (t_1,t_2)$, $\dot{R}\geq 0$.
I can proof in the left interval, $y(t)$ is bounded. here is my attempt:
$$ \frac{d}{dt}(R y^2 + \dot{y}^2) = 2\dot{y}(\ddot{y}+R y^2) + \dot{R} y^2 \leq 0 $$
then $$ R y^2 \leq \dot{y}^2 + R y^2 \leq \dot{y}(t_0)^2 + R y(t_0)^2 = M_0$$
so $$y\leq \sqrt{\frac{M_0}{R}}$$
how to get the right interval result, get the upper boundary of $y$?
Is there a more general result for $R\geq 0 $? ie. $y$ has upper and lower bounds .