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

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