A Maple package to find first order differential invariants of 2ODEs via a Darboux approach

Here we present an implementation of a semi-algorithm to find elementary first order differential invariants (elementary first integrals) of a class of rational second order ordinary differential equations (rational 2ODEs). The algorithm was developed in Duarte and da Mota (2009)  [18]; it is based on a Darboux-type procedure, and it is an attempt to construct an analog (generalization) of the method built by Prelle and Singer (1983)  [6] for rational first order ordinary differential equations (rational 1ODEs). to deal, this time, with 2ODEs. The FiOrDi package presents a set of software routines in Maple for dealing with rational 2ODEs. The package presents commands permitting research investigations of some algebraic properties of the ODE that is being studied.Program summaryProgram title: FiOrDiCatalogue identifier: AEQL_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEQL_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 2262No. of bytes in distributed program, including test data, etc.: 83 609Distribution format: tar.gzProgramming language: Maple (release 14).Computer: PC.Operating system: Windows 7.RAM: 128 MBClassification: 4.3, 5.Nature of problem:Determination of first order differential invariants for rational second order ordinary differential equations.Solution method:The method of solution is based on a Darboux/PS-type approachRestrictions:If, for the ODE under consideration, the Darboux polynomials are of high degree (>3) in the dependent and independent variables, the package may spend an impractical amount of time to obtain the solution.Unusual features:Our implementation not only searches for differential first order invariants, but can also be used as a research tool that allows the user to follow all the steps of the procedure (for example, we can calculate the associated “DD” operator, the corresponding Darboux polynomials, and associated co-factors, etc.). In addition, since our package is based on recent theoretical developments [1], it can successfully reduce some rational 2ODEs that were not solved (or reduced) by some of the best-known methods available.Running time:This depends strongly on the ODE, but usually under 4 s.References:[1]L.G.S. Duarte, L.A.C.P. da Mota, Finding elementary first integrals for rational second order ordinary differential equations, Journal of Mathematical Physics 50 (1) (2009) 013514–013514-17.