Computer Algebra Solving of First Order ODEs Using Symmetry Methods

A set of Maple V R.3/4 computer algebra routines for the analytical solving of 1st. order ODEs, using Lie group symmetry methods, is presented. The set of commands includes a 1st. order ODE-solver and routines for, among other things: the explicit determination of the coefficients of the infinitesimal symmetry generator; the construction of the most general invariant 1st. order ODE under given symmetries; the determination of the canonical coordinates of the underlying invariant group; and the testing of the returned results.Comment: 14 pages, LaTeX, submitted to Computer Physics Communications. Soft-package (On-Line Help) and sample MapleV session available at: http://dft.if.uerj.br/symbcomp.htm or ftp://dft.if.uerj.br/pdetool

Computer Algebra Solving of Second Order ODEs Using Symmetry Methods

An update of the ODEtools Maple package, for the analytical solving of 1st and 2nd order ODEs using Lie group symmetry methods, is presented. The set of routines includes an ODE-solver and user-level commands realizing most of the relevant steps of the symmetry scheme. The package also includes commands for testing the returned results, and for classifying 1st and 2nd order ODEs.Comment: 24 pages, LaTeX, Soft-package (On-Line help) and sample MapleV sessions available at: http://dft.if.uerj.br/odetools.htm or http://lie.uwaterloo.ca/odetools.ht

Integrating Factors for Second-order ODEs

AbstractA systematic algorithm for building integrating factors of the form μ(x,y), μ(x,y′) or μ(y,y′) for second-order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the μ(x,y) problem. Examples of ODEs not having point symmetries are shown to be solvable using this algorithm. The scheme was implemented in Maple, in the framework of the ODEtools package and its ODE-solver. A comparison between this implementation and other computer algebra ODE-solvers in tackling non-linear examples from Kamke's book is shown

Symmetries and First Order ODE Patterns

A scheme for determining symmetries for certain families of rst order ODEs, without solving any dierential equations, and based mainly in matching an ODE to patterns of invariant ODE families, is presented. The scheme was implemented in Maple, in the framework of the ODEtools package and its ODE-solver. A statistics of the performance of this approach in solving the rst order ODE examples of Kamke's book [1] is shown. (Revised Version. To appear in Computer Physics Communications) 1 Department of Mathematics, University of British Columbia, Vancouver, Canada. 2 Symbolic Computation Group, Department of Theoretical Physics, State University of Rio de Janeiro, Brasil. 3 Department of Computer Science, Faculty of Mathematics, University of Waterloo, Ontario, Canada. Available as http://dft.if.uerj.br/preprint/e8-1.tex; also as http://lie.uwaterloo.ca/odetools/ode iv.tex PROGRAM SUMMARY Title of the software package: Extension to the Maple ODEtools package Catalogue number: (sup..