Comments on “investigation of the properties of the period for the nonlinear oscillator x¨+(1+x˙2)x=0”

Two Lindstedt–Poinaré perturbation-based methods are used to solve the nonlinear differential equation of a nonlinear oscillator having the square of the angular frequency quadratic dependence on the velocity. Mickens published two interesting papers [J. Beatty, R.E. Mickens, A qualitative study of the solutions to the differential equation x¨+(1+x˙2)x=0, Journal of Sound and Vibration   283 (2005) 475–477; R.E. Mickens, Investigation of the properties of the period for the nonlinear oscillator x¨+(1+x˙2)x=0, Journal of Sound and Vibration 292 (2006) 1031–1035] about this oscillator and by using the harmonic balance method he found that the approximate frequency is not defined for amplitudes of magnitude equal to or larger than two. We show that these standard perturbation methods work better than the harmonic balance method. In particular, the modified Lindstedt–Poincaré method works well for the whole range of oscillation amplitudes, and excellent agreement of the approximate frequency with the exact one has been demonstrated and discussed.