Exact solutions of the cubic-quintic Duffing equation using leaf functions

The exact solutions of both the cubic Duffing equation and cubic-quintic are presented by using only leaf functions. In previous studies, have been proposed functions that integrate in phase trigonometric Because they not simple, procedures for transforming complicated convenient. first derivative function can be derived as root. This factored. These factors or multiplications to equation. Some these same type reported previously. satisfy cubic--quintic equations with high nonlinearity. this study, relationship between parameters coefficients terms is clarified. We numerically analyze solutions, plot waveform, discuss periodicity amplitude waveform.