Open Access
2013 Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry
K. S. Mahomed, E. Momoniat
Abstr. Appl. Anal. 2013(SI06): 1-8 (2013). 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 equation y ′′′ = 0 is 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.

Citation

Download Citation

K. S. Mahomed. E. Momoniat. "Algebraic Properties of First Integrals for Scalar Linear Third-Order ODEs of Maximal Symmetry." Abstr. Appl. Anal. 2013 (SI06) 1 - 8, 2013. https://doi.org/10.1155/2013/530365

Information

Published: 2013
First available in Project Euclid: 26 February 2014

zbMATH: 1274.34104
MathSciNet: MR3035207
Digital Object Identifier: 10.1155/2013/530365

Rights: Copyright © 2013 Hindawi

Vol.2013 • No. SI06 • 2013
Back to Top