  According to the wave power rule, the second derivative of a function with
respect to the variable t is equal to negative n times the function raised to
the power of 2n-1. Solving the ordinary differential equations numerically
results in waves appearing in the figures. The ordinary differential equation
is very simple; however, waves, including the regular amplitude and period, are
drawn in the figure. In this study, the function for obtaining the wave is
called the leaf function. Based on the leaf function, the exact solutions for
the undamped and unforced Duffing equations are presented. In the ordinary
differential equation, in the positive region of the variable, the second
derivative becomes negative. Therefore, in the case that the curves vary with
the time under the condition x(t)>0, the gradient constantly decreases as time
increases. That is, the tangential vector on the curve of the graph changes
from the upper right direction to the lower right direction as time increases.
On the other hand, in the negative region of the variable, the second
derivative becomes positive. The gradient constantly increases as time
decreases. That is, the tangent vector on the curve changes from the lower
right direction to the upper right direction as time increases. Since the
behavior occurring in the positive region of the variable and the behavior
occurring in the negative region of the variable alternately occur in regular
intervals, waves appear by these interactions. In this paper, I present seven
types of damped and divergence exact solutions by combining trigonometric
functions, hyperbolic functions, hyperbolic leaf functions, leaf functions, and
exponential functions. In each type, I show the derivation method and numerical
examples, as well as describe the features of the waveform.
