A classification, according to invariant theory, of non-constant invariant Abel ODEs known as solvable and found in the literature is presented. A set of new integrable classes depending on one or no parameters, derived from the analysis of the works by Abel, Liouville and Appell [2,3,4], is also shown. Computer algebra routines were developed to solve ODEs members of these classes by solving their related equivalence problem. The resulting library permits a systematic solving of Abel type ODEs in the Maple symbolic computing environment. Keywords: Abel type first order ordinary differential equations (ODEs), equivalence problem, integrable cases, symbolic computation. PROGRAM SUMMARY Nature of mathematical problemAnalytical solving of Abel type first order ODEs having non-constant invariant. Methods of solutionSolving the equivalence problem between a given ODE and representatives of a set of non-constant invariant Abel ODE classes for which solutions are available.Restrictions concerning the complexity of the problem The computational routines presented work when the input ODE belongs to one of the Abel classes considered in this work. This set of Abel classes can be extended, but there are classes -depending intrinsically on many parameters -for which the solution of the equivalence problem, as presented here, may lead to large and therefore untractable expressions. When the invariants of a given Abel ODE depend on analytic functions, the success of the routines depends on Maple's ability to normalize these invariants and recognize zeros (this is well implemented in Maple, but it may nevertheless not work as expected in some cases). Also, when the solution for the class parameter depends on other algebraic symbols entering the ODE being solved, the routines can determine this dependency only when it has rational form. Typical running timeThe methods being presented here have been implemented in the framework of the ODEtools Maple package. On the average, over Kamke's [1] first order Abel examples (see sec. 6), the ODE-solver of ODEtools is now spending ≈ 6 sec. per ODE when successful, and ≈ 11 sec. when unsuccessful. The timings in this paper were obtained using Maple R5 on a Pentium II 400 -128 Mb. of RAM -running Windows98.Unusual features of the program These computational routines are able -in principle -to integrate the infinitely many members of all the nonconstant invariant Abel ODE classes considered in this work. Concretely, when a given Abel ODE belongs to one of these classes, the routines can determine this fact, by solving the related equivalence problem, and then use that information to return a closed form solution without requiring further participation from the user. The ODE families that are covered include, as particular cases, all the Abel solvable cases presented in Kamke's and Murphy's books, as well as the Abel ODEs member of other classes not previously presented in the literature to the best of our knowledge. After incorporating the new routines, the ODE solver of the ODEtoo...

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 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.