  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 sleaf2(l) can be derived as the root of 1-(sleaf2(l))^4. This
derivative can be factored, and the factors are 1+sleaf2(l), 1-sleaf2(l), and
1+(sleaf2(l))^2. 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. These rules can also be applied to the leaf
function cleaf2(l). 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.
