Dealing with rational second order ordinary differential equations where both Darboux and Lie find it difficult: The [formula omitted]-function method

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Here we present a new approach to search for first order invariants (first integrals) of rational second order ordinary differential equations. This method is an alternative to the Darbouxian and symmetry approaches. Our procedure can succeed in many cases where these two approaches fail. We also present here a Maple implementation of the theoretical results and methods, hereby introduced, in a computational package — InSyDE. The package is designed, apart from materializing the algorithms presented, to provide a set of tools to allow the user to analyse the intermediary steps of the process. Computer programs in physicsProgram Title: InSyDE — Invariants and Symmetries of (racional second order ordinary) Differential Equations.Program Files doi:http://dx.doi.org/10.17632/4ytft6zgk7.1Licensing provisions: GPLv3Programming language: Maple 17.Nature of problemSearch for first integrals of rational 2ODEs.Solution method:The method of solution is based on an algorithm described in this paper.Restrictions:If the rational 2ODE that is being analysed presents a very high degree in (x,y,z), then the method may not work well.Unusual features:Our implementation can find first integrals in many cases where the rational 2ODE under study cannot be reduced by other powerful solvers. Besides that, the package presents some useful research commands.