Exact Solutions of the Cubic-Quintic Duffing Equation Using Leaf Functions

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