Integration of Third Order Ordinary Differential Equations Possessing Two-Parameter Symmetry Group by Lie's Method

TL;DR

This paper develops a Lie group theoretic method to solve third order ordinary differential equations with two-parameter symmetry groups, focusing on reduction techniques involving hidden symmetries and symmetry preservation.

Contribution

It introduces a systematic approach for reducing third order ODEs with two-parameter symmetries using symmetry analysis and hidden symmetry detection.

Findings

Reduction to quadratures is possible via symmetry analysis.

Identification of hidden symmetries aids in reduction.

Constructed reduction paths for higher order equations.

Abstract

The solution of a class of third order ordinary differential equations possessing two parameter Lie symmetry group is obtained by group theoretic means. It is shown that reduction to quadratures is possible according to two scenarios: 1) if upon first reduction of order the obtained second order ordinary differential equation besides the inherited point symmetry acquires at least one more new point symmetry (possibly a hidden symmetry of Type II). 2) First, reduction paths of the fourth order differential equations with four parameter symmetry group leading to the first order equation possessing one known (inherited) symmetry are constructed. Then, reduction paths along which a third order equation possessing two-parameter symmetry group appears are singled out and followed until a first order equation possessing one known (inherited) symmetry are obtained. The method uses conditions for preservation, disappearance and reappearance of point symmetries.