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Mathematical Problems in Engineering
Volume 2013, Article ID 214872, 9 pages
Research Article

Characterization of Symmetry Properties of First Integrals for Submaximal Linearizable Third-Order ODEs

Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, South Africa

Received 23 August 2013; Accepted 13 September 2013

Academic Editor: Chaudry Masood Khalique

Copyright © 2013 K. S. Mahomed and E. Momoniat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The relationship between first integrals of submaximal linearizable third-order ordinary differential equations (ODEs) and their symmetries is investigated. We obtain the classifying relations between the symmetries and the first integral for submaximal cases of linear third-order ODEs. It is known that the maximum Lie algebra of the first integral is achieved for the simplest equation and is four-dimensional. We show that for the other two classes they are not unique. We also obtain counting theorems of the symmetry properties of the first integrals for these classes of linear third-order ODEs. For the 5 symmetry class of linear third-order ODEs, the first integrals can have 0, 1, 2, and 3 symmetries, and for the 4 symmetry class of linear third-order ODEs, they are 0, 1, and 2 symmetries, respectively. In the case of submaximal linear higher-order ODEs, we show that their full Lie algebras can be generated by the subalgebras of certain basic integrals.