Solving first order differential equations presenting elementary functions

TL;DR

This paper develops a theoretical framework linking elementary function-based first order differential equations to rational second order equations with Liouvillian integrals, enabling efficient integration and analysis of chaotic systems.

Contribution

It establishes a new theoretical connection between 1ODEs with elementary functions and rational 2ODEs, expanding the S-function method for solving complex differential equations.

Findings

Method effectively integrates 1ODEs with elementary functions.

Algorithm can identify regions of integrability in chaotic systems.

Theoretical basis enhances understanding of differential equations with Liouvillian integrals.

Abstract

We have already dealt with the problem of solving First Order Differential Equations (1ODEs) presenting elementary functions before in [1, 2]. In this present paper, we have established solid theoretical basis through a relation between the 1ODE we are dealing with and a rational second order ordinary differential equation, presenting a Liouvillian first Integral. Here, we have expanded the results in [3], where we have establish a theoretical background to deal with rational second order ordinary differential equations (2ODEs) via the S-function method. Using this generalisation and other results hereby introduced, we have produced a method to integrate the 1ODE under scrutiny. Our methods and algorithm are capable to deal efficiently with chaotic systems, determining regions of integrability.