Weak periodic solutions of $x\ddot{x} + 1 = 0$ and the harmonic balance method

We prove that the differential equation $x\ddot{x} + 1 = 0$ has continuous weak periodic solutions and compute their periods. Then, we use the Harmonic Balance Method until order six to approach these periods and to illustrate how the sharpness of the method increases with the order. Our computations rely on the Gr\"obner basis method.