Integrating factors for second order ODEs

TL;DR

This paper presents a systematic algorithm for finding integrating factors of second order ODEs, enabling their solution without solving additional differential equations, and demonstrates its implementation in Maple.

Contribution

The paper introduces a new algorithm that determines integrating factors for second order ODEs without solving extra equations, including for equations lacking point symmetries.

Findings

Algorithm successfully finds integrating factors for various second order ODEs.

Implementation in Maple's ODEtools enhances the solving capabilities for complex ODEs.

Comparison shows improved performance over existing solvers on non-linear examples.

Abstract

A systematic algorithm for building integrating factors of the form mu(x,y), mu(x,y') or mu(y,y') for second order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the mu(x,y) problem. Examples of ODEs not having point symmetries are shown to be solvable using this algorithm. The scheme was implemented in Maple, in the framework of the "ODEtools" package and its ODE-solver. A comparison between this implementation and other computer algebra ODE-solvers in tackling non-linear examples from Kamke's book is shown.