Symmetry Classification of First Integrals for Scalar Linearizable Second-Order ODEs

Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations (ODEs) which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0-, 1-, 2-, or 3-point symmetry cases. It is shown that the maximal algebra case is unique.

[1]  Fazal M. Mahomed,et al.  Symmetry Lie algebras of nth order ordinary differential equations , 1990 .

[2]  A. Kara,et al.  A Basis of Conservation Laws for Partial Differential Equations , 2002 .

[3]  S. Lie,et al.  Classification und Integration von gewhnlichen Differentialgleichungen zwischenxy, die eine Gruppe von Transformationen gestatten: Die nachstehende Arbeit erschien zum ersten Male im Frhling 1883 im norwegischen Archiv , 1888 .

[4]  Fazal M. Mahomed,et al.  Lie and Noether Counting Theorems for One-Dimensional Systems , 1993 .

[5]  Fazal M. Mahomed,et al.  Lie algebras associated with scalar second-order ordinary differential equations , 1989 .

[6]  Fazal M. Mahomed,et al.  Symmetry group classification of ordinary differential equations: Survey of some results , 2007 .

[7]  K. Govinder,et al.  The Algebraic Structure of the First Integrals of Third-Order Linear Equations , 1995 .

[8]  P. Winternitz,et al.  Ordinary di ff erential and di ff erence equations invariant under SL ( 2 , R ) and their solutions , 2009 .

[9]  F. Mahomed,et al.  Maximal subalgebra associated with a first integral of a system possessing sl(3,R) algebra , 1988 .

[10]  S. Lie.  Classification und Integration von gewöhnlichen Differentialgleichungen zwischenxy, die eine Gruppe von Transformationen gestatten , 1888 .

[11]  Ar. Tresse,et al.  Sur les invariants différentiels des groupes continus de transformations , 1894 .

[12]  F. Mahomed,et al.  THE LIE ALGEBRA sl(3, R) AND LINEARIZATION , 1989 .

[13]  F. Mahomed,et al.  THE LINEAR SYMTRIES OF A NONLINEAR DIFFERENTIAL EQUATION , 1985 .

[14]  P. Hartman.  Ordinary Differential Equations , 1965 .