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Manuscript Title: Abel ODEs: equivalence and integrable classes.
Authors: E.S. Cheb-Terrab, A.D. Roche
Program title: Extension to ODEtools package
Catalogue identifier: ADMB_v1_0
Distribution format: zip
Journal reference: Comput. Phys. Commun. 130(2000)204
Programming language: Maple V Release 4 and 5.
Computer: Pentium II 400 -128Mb.
Operating system: UNIX, Macintosh, Windows (95/98/NT).
RAM: 16M words
Keywords: Computer algebra, Abel type first order ordinary differential equations (ODEs), Equivalence problem, Integrable cases, Symbolic computation.
Classification: 5.

Nature of problem:
Analytical solving of Abel type first order ODEs having non-constant invariant.

Solution method:
Solving the equivalence problem between a given ODE and representatives of a set of non-constant invariant Abel ODE classes for which solutions are avaliable.

Restrictions:
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.

Unusual features:
These computational routines are able - in principle - to integrate the infinitely many members of all the non-constant 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 ODEtools package succeeds in solving 97 per cent of Kamke's first order examples.

Running time:
The 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.

References:
[1] E. Kamke, Differentialgleichungen: Losungsmethoden und Losungen, (Chelsea, New York, 1959).