I've got another question to pose to you. I am given the differential equation $$x''(t)+a(t)f(x(t))=0,$$ with $a(t)\geq 1$, $f\geq 0$, $$ \int_0^{+\infty} f(y)\mathrm d y=+\infty$$ and $$a,f\in C^0(\mathbb R).$$ Then set $I=(t_0,t_1)$ be the maximal interval of definition of the solution; I am then asked to prove that $x(t)$ is bounded above as $t\to t_1^-$. Thanks in advance for your courtesy.
Regards