An extension of the ideas of the Prelle-Singer procedure to second order differential equations is proposed. As in the original PS procedure, this version of our method deals with differential equations of the form y ′′ = M (x, y, y ′ )/N (x, y, y ′ ), where M and N are polynomials with coefficients in the field of complex numbers C . The key to our approach is to focus not on the final solution but on the first-order invariants of the equation. Our method is an attempt to address algorithmically the solution of SOODEs whose first integrals are elementary functions of x, y and y ′ .

An update of the ODEtools Maple package, for the analytical solving of 1 st and 2 nd 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 1 st and 2 nd order ODEs. PROGRAM SUMMARYTitle of the software package: ODEtools. Catalogue number: (supplied by Elsevier)Software obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland (see application form in this issue) Licensing provisions: noneOperating systems under which the program has been tested: UNIX systems, Macintosh, DOS (AT 386, 486 and Pentium based) systems, DEC VMS, IBM CMS.Programming language used: Maple V Release 3/4. Memory required to execute with typical data: 16 Megabytes.No. of lines in distributed program, excluding On-Line Help, etc.: 10159.Keywords: 1 st /2 nd order ordinary differential equations, symmetry methods, symbolic computation. Nature of mathematical problemAnalytical solving of 1 st and 2 nd order ordinary differential equations using symmetry methods, and the inverse problem; that is: given a set of point and/or dynamical symmetries, to find the most general invariant 1 st or 2 nd order ODE. Methods of solutionComputer algebra implementation of Lie group symmetry methods. Restrictions concerning the complexity of the problemBesides the inherent restrictions of the method (there is as yet no general scheme for solving the associated PDE for the coefficients of the infinitesimal symmetry generator), the present implementation does not work with systems of ODEs nor with ODEs of differential order higher than two. Typical running timeThis depends strongly on the ODE to be solved. For the case of first order ODEs, it usually takes from a few seconds to 1 or 2 minutes. In the tests we ran with the first 500 1 st order ODEs from Kamke's book [1], the average times were: 8 sec. for a solved ODE and 15 sec. for an unsolved one, using a Pentium 200 with 64 Mb. RAM, on a Windows 95 platform. In the case of second order ODEs, the average times for the non-linear 2 nd order examples of Kamke's Book were 35 seconds for a solved ODE and 50 seconds for an unsolved one. The tests were run using the Maple version under development, but almost equivalent results are obtained using the available Maple R4 and R3 (the code presented in this work runs in all these versions). Unusual features of the programThe ODE-solver here presented is an implementation of all the steps of the symmetry method solving scheme; that is, the command receives an ODE, and when successful it directly returns a closed form solution for the undetermined function. Also, this solver permits the user to optionally participate in the solving process by giving advice concerning the functional form for the coefficients of the infinitesimal symmetry generator (infinitesimals). Many of the intermediate steps of the symmetry scheme are available as ...

A set of Maple V R.3/4 computer algebra routines for the analytical solving of 1 st order ODEs, using Lie group symmetry methods, is presented. The set of commands includes a 1 st order ODEsolver and routines for, among other things: the explicit determination of the coefficients of the infinitesimal symmetry generator; the construction of the most general invariant 1 st order ODE under given symmetries; the determination of the canonical coordinates of the underlying invariant group; and the testing of the returned results. PROGRAM SUMMARYTitle of the software package: First order ODE tools. Catalogue number: (supplied by Elsevier)Software obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland (see application form in this issue) Licensing provisions: noneOperating systems under which the program has been tested: UNIX systems, Macintosh, DOS (AT 386, 486 and Pentium based) systems, DEC VMS, IBM CMS.Programming language used: Maple V Release 3/4. Memory required to execute with typical data: 8 Megabytes.No. of lines in distributed program, including On-Line Help, etc.: 4183.Keywords: 1 st order ordinary differential equations, symmetry methods, symbolic computation. Nature of mathematical problemAnalytical solving of 1 st order ordinary differential equations. Methods of solutionComputer algebra implementation of Lie group symmetry methods. Restrictions concerning the complexity of the problemBesides the inherent restrictions of the method (there is as yet no general scheme for solving the associated PDE for the coefficients of the infinitesimal symmetry generator), the present implementation does not work with systems of ODEs nor with higher order ODEs. Typical running timeThis depends strongly on the ODE to be solved, usually taking from a few seconds to a few minutes. In the tests we ran (with 466 1 st order ODEs from Kamke's book [5], see sec.4), the average times were: 6 sec. for a solved ODE and 20 sec. for an unsolved one, using a Pentium 133 with 64 Mb. RAM on a Windows 3.11 platform. Unusual features of the programThe 1 st order ODE-solver here presented is an implementation of all the steps of the symmetry method solving scheme; i.e., when successful it returns a closed solution, not the symmetry generator. Also, this solver permits the user to (optionally) participate in the solving process by giving an advice (HINT option) concerning the functional form for the coefficients of the infinitesimal symmetry generator (infinitesimals). All the intermediate steps of the symmetry method solving scheme are available as user-level commands too. For instance, using the package's commands, it is possible to obtain the infinitesimals, the related canonical coordinates, and the most general 1 st order ODE invariant under a symmetry group. The package also includes a command for classifying ODEs (according to Kamke's' book[5]) popping up Help pages with Kamke's advice for solving them, facilitating the study of a given ODE and the use of the package with pedagogical purposes.

Here we present a semialgorithm to find elementary first integrals of a class of rational second order ordinary differential equations. The method is based on a Darboux-type procedure and it is an attempt to construct an analogous of the method built by Prelle and Singer [“Elementary first integral of differential equations,” Trans. Am. Math. Soc. 279, 215 (1983)] for rational first order ordinary differential equations.

We present an algorithm to solve First Order Ordinary Differential Equations (FOODEs) extending the Prelle-Singer (PS) Method. The usual PS-approach miss many FOODEs presenting Liouvillian functions in the solution (LFOODEs). We point out why and propose an algorithm to solve a large class of these previously unsolved LFOODEs. Although our algorithm does not cover all the LFOODEs, it is an elegant extension mantaining the semi-decision nature of the usual PS-Method.