Can I use Sturm's comparison theorem to say something about the average?
Let $f(t)$ be a continuous function such that $\lim_{t \to \infty } f(t) = \infty$. Let us consider the following function
$$ \ddot{x}(t) + t^2 x(t) = 0. $$
Let $C$ be a constant. Then, we can find $T$ such that $C^2 < t^2$ for all $t > T$. By Sturm's comparison theorem, exist $t^*$ in $(\frac{2\pi k}{C},\frac{2\pi (k+1)}{C})$ such that $x(t^*) = 0$ for every $k$ big enough.
Can we concluded the following?
$$ \lim_{t \to \infty} x(t) =0 $$
or
$$ \lim_{t \to \infty} \frac{1}{t}\int_0^t s x(s) ds =0 $$