JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 212, 349]374Ž. 1997 ARTICLE NO. AY975506 Characterisation of the Algebraic Properties of First Integrals of Scalar Ordinary Differential Equations of Maximal Symmetry G. P. Flessas,* K. S. Govinder,² and P. G. L. Leach³ View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Department of Mathematics, Uni¨ersity of the Aegean, Karlo¨assi, 83 200, Samos, Greece Submitted by Thanasis Fokas Received March 10, 1995 We undertake a study of the first integrals of linear nth order scalar ordinary differential equations with maximal symmetry. We establish patterns for the first integrals associated with these equations. It is shown that second and third order equations are the pathological cases in the study of higher order differential equations. The equivalence of contact symmetries for third order equations to non-Cartan symmetries of second order equations is highlighted. Q 1997 Academic Press 1. INTRODUCTION A scalar ordinary differential equation Žn. ExŽ.,y,y9,..., y s0,Ž. 1 *E-mail address:
[email protected]. ² Permanent address: Department of Mathematics and Applied Mathematics, University of Natal, Durban, 4041, Republic of South Africa. E-mail address:
[email protected]. ³ Permanent address: Department of Mathematics and Applied Mathematics, University of Natal, Durban, 4041, Republic of South Africa. Member of the Centre for Theoretical and Computational Chemistry, University of Natal and associate member of the Centre for Nonlinear Studies, University of the Witwatersrand, Johannesburg, Republic of South Africa. E-mail address:
[email protected].