Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 171834, 17 pages
http://dx.doi.org/10.1155/2011/171834
Research Article

Linearizability of Systems of Ordinary Differential Equations Obtained by Complex Symmetry Analysis

M. Safdar,1 Asghar Qadir,1 and S. Ali2,3

1Center for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus H-12, Islamabad 44000, Pakistan
2School of Electrical Engineering and Computer Science, National University of Sciences and Technology, Campus H-12, Islamabad 44000, Pakistan
3Department of Mathematics, Brock University, St. Catherines, ON, Canada L2S 3A1

Received 16 April 2011; Accepted 22 August 2011

Academic Editor: Dane Quinn

Copyright © 2011 M. Safdar et al. 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.

Abstract

Five equivalence classes had been found for systems of two second-order ordinary differential equations, transformable to linear equations (linearizable systems) by a change of variables. An “optimal (or simplest) canonical form” of linear systems had been established to obtain the symmetry structure, namely, with 5-, 6-, 7-, 8-, and 15-dimensional Lie algebras. For those systems that arise from a scalar complex second-order ordinary differential equation, treated as a pair of real ordinary differential equations, we provide a “reduced optimal canonical form.” This form yields three of the five equivalence classes of linearizable systems of two dimensions. We show that there exist 6-, 7-, and 15-dimensional algebras for these systems and illustrate our results with examples.