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According to the wave power rule, the second derivative of a function withrespect to the variable t is equal to negative n times the function raised tothe power of 2n-1. Solving the ordinary differential equations numericallyresults in waves appearing in the figures. The ordinary differential equationis very simple; however, waves, including the regular amplitude and period, aredrawn in the figure. In this study, the function for obtaining the wave iscalled the leaf function. Based on the leaf function, the exact solutions forthe undamped and unforced Duffing equations are presented. In the ordinarydifferential equation, in the positive region of the variable, the secondderivative becomes negative. Therefore, in the case that the curves vary withthe time under the condition x(t)>0, the gradient constantly decreases as timeincreases. That is, the tangential vector on the curve of the graph changesfrom the upper right direction to the lower right direction as time increases.On the other hand, in the negative region of the variable, the secondderivative becomes positive. The gradient constantly increases as timedecreases. That is, the tangent vector on the curve changes from the lowerright direction to the upper right direction as time increases. Since thebehavior occurring in the positive region of the variable and the behavioroccurring in the negative region of the variable alternately occur in regularintervals, waves appear by these interactions. In this paper, I present seventypes of damped and divergence exact solutions by combining trigonometricfunctions, hyperbolic functions, hyperbolic leaf functions, leaf functions, andexponential functions. In each type, I show the derivation method and numericalexamples, as well as describe the features of the waveform.

Damped and Divergence Exact Solutions for the Duffing Equation using Leaf Functions and Hyperbolic Leaf Functions