Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry

K S Mahomed,E Momoniat

doi:10.1155/2013/530365

Abstract

By use of the Lie symmetry group methods we analyze the relationship between the first integrals of the simplest linear third-order ordinary differential equations (ODEs) and their point symmetries. It is well known that there are three classes of linear third-order ODEs for maximal cases of point symmetries which are 4, 5, and 7. The simplest scalar linear third-order equation has seven-point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equationy′′′=0is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals.

Highlights

Ordinary differential equations (ODEs) are a fertile area of study, especially the Lie algebraic properties of such equations and their first integrals have great importance
We show that only X = ξ(x, y)∂/∂x are symmetries of the first integral
We cannot generate the full algebra as is the case for linear scalar second-order ODEs [11] by use of the algebra of the basic integral alone

Summary

Research Article

Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry. By use of the Lie symmetry group methods we analyze the relationship between the first integrals of the simplest linear third-order ordinary differential equations (ODEs) and their point symmetries. The simplest scalar linear third-order equation has seven-point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. We show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs. we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals

Introduction

It is easy to see that

These are obtained by setting

Some Remarks

Yi xi