SCHWEITZER 1975 Problem 9

Let $l_0,c,\alpha,g$ be positive constants, and let $x(t)$ be the solution of the differential equation $$([l_0+ct^{\alpha}] ^2x')'+g[l_0+ct^{\alpha}] \sin x=0, t \geq 0,\ \;-\frac{\pi}{2} <x< \frac{\pi}{2},$$ satisfying the initial conditions $x(t_0)=x_0, x'(t_0)=0$ . (This is the equation of the mathematical pendulum whose length changes according to the law $l=l_0+ct^{\alpha}$ .) Prove that $x(t)$ is defined on the interval $[t_0,\infty)$ ; furthermore, if $\alpha >2$ then for every $x_0 \not= 0$ there exists a $t_0$ such that $$\liminf_{t \rightarrow \infty} |x(t)| >0.$$

L. Hatvani