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

## TL;DR

This paper derives exact solutions for the Duffing equation using leaf functions and hyperbolic leaf functions, revealing wave behaviors and presenting seven solution types with numerical examples.

## Contribution

It introduces novel exact solutions for the Duffing equation based on leaf functions, expanding analytical methods for nonlinear differential equations.

## Key findings

- Seven types of damped and divergence solutions are derived.
- Waveform features are characterized through numerical examples.
- Solutions combine trigonometric, hyperbolic, and exponential functions.

## Abstract

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.