Preliminary Analysis of the Inverse-x Oscillator

The inverse-x oscillator has the following equation \begin{equation} \ddot{x}+\frac{1}{x}=0. \end{equation} While the motion is singular at $x=0$, nevertheless, all of the solutions are bounded in $x$ and periodic in time. In addition to proving these behaviors, we calculate the exact period and present a (preliminary) analysis of the periodic solutions with respect to the construction of analytic approximations. The procedures used involve phase-space methods and harmonic balance.