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A semi-algorithm to find elementary first order invariants of rational second order ordinary differential equations AbstractHere we present a method to find elementary first integrals of rational second order ordinary differential equations (SOODEs) based on a Darboux type procedure [16,12,13]. Apart from practical computational considerations, the method will be capable of telling us (up to a certain polynomial degree) if the SOODE has an elementary first integral and, in positive case, finds it via quadratures.

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. PROGRAM SUMMARYTitle of the software package: InSyDE -Invariants and Symmetries of (racional second order ordinary) Differential Equations.Catalogue number: (supplied by Elsevier) Software obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland. Licensing provisions: noneOperating systems under which the program has been tested: Windows 8. Programming languages used: Maple 17.Memory required to execute with typical data: 200 Megabytes. No. of lines in distributed program, including On-Line Help, etc.: 537Nature of mathematical problem Search for first integrals of rational 2ODEs. Methods of solutionThe method of solution is based on an algorithm described in this paper. Restrictions concerning the complexity of the problemIf the rational 2ODE that is being analysed presents a very high degree in (x, y, z), then the method may not work well. Typical running timeThis depends strongly on the 2ODE that is being analysed. Unusual features of the programOur implementation can find first integrals in many cases where the rational 2ODE under study can not be reduced by other powerful solvers. Besides that, the package presents some useful research commands.

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Here we present/implement an algorithm to find Liouvillian first integrals of dynamical systems in the plane. In [1], we have introduced the basis for the present implementation. The particular form of such systems allows reducing it to a single rational first order ordinary differential equation (rational first order ODE). We present a set of software routines in Maple 10 for solving rational first order ODEs. The package present commands permitting research incursions of some algebraic properties of the system that is being studied.Keyword: Liouvillian functions, first integrals, dynamical systems in the plane, first order ordinary differential equations, computer algebra, Prelle-Singer (PS) PROGRAM SUMMARYTitle of the software package: Lsolver.Catalogue number: (supplied by Elsevier) Software obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland. Licensing provisions: noneOperating systems under which the program has been tested: Windows ME, Windows XP. Programming languages used: Maple 10Memory required to execute with typical data: 128 Megabytes.No. of lines in distributed program, including On-Line Help, etc.: 900.Keywords: Liouvillian functions, first integrals, dynamical systems in the plane, first order ordinary differential equations, computer algebra, Prelle-Singer (PS). Nature of mathematical problemSolution of rational first order ordinary differential equations. Methods of solutionThe method of solution is based on a Darboux/PS type approach.Restrictions concerning the complexity of the problem If the integrating factor for the rational first order ODE under consideration presents Darboux Polynomials of high degree ( > 3 ) in the dependent and independent variables, the package may spend an unpractical amount of time to obtain the solution. Typical running timeThis depends strongly on the ODE, but usually under 2 seconds. Unusual features of the programOur implementation not only search for Liouvillian conserved quantities, but can also be used as a research tool that allows the user to follow all the steps of the Darboux procedure (for example, the algebraic invariants curves and associated cofactors can be calculated). In addition, since our package is based in recent theoretical developments [1], it can successfully solve a class of rational first order ODEs that were not solved by some of the best-known ODE solvers available.

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