Solution of a mixed parity nonlinear oscillator: Harmonic balance

The method of harmonic balance is used to calculate first-order approximations to the periodic solutions of a mixed parity nonlinear oscillator. First, the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction. Then the two auxiliary equations, where the restoring force functions are odd, are solved by using a first harmonic term (without a constant). Therefore, we obtain the two approximate solutions to the mixed parity nonlinear oscillator. One is expressed in terms of the exact amplitude in the negative direction, the other in terms of the approximate amplitude. These solutions are more accurate than the second approximate solution obtained by the Lindstedt–Poincaré method for large amplitudes.