Harmonic balance approaches to the nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable

The second-order harmonic balance method is used to construct three approximate frequency–amplitude relations for a conservative nonlinear singular oscillator in which the restoring force is inversely proportional to the dependent variable. Two procedures are used to solve the nonlinear differential equation approximately. In the first the differential equation is rewritten in a form that does not contain the y−1 expression, while in the second the differential equation is solved directly. The approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first one and the discrepancy between the approximate frequency and the exact one is lower than 1.28%.