  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.
