PSsolver: A Maple implementation to solve first order ordinary differential equations with Liouvillian solutions

Abstract

We present a set of software routines in Maple 14 for solving first order ordinary differential equations (FOODEs). The package implements the Prelle–Singer method in its original form together with its extension to include integrating factors in terms of elementary functions. The package also presents a theoretical extension to deal with all FOODEs presenting Liouvillian solutions. Applications to ODEs taken from standard references show that it solves ODEs which remain unsolved using Maple’s standard ODE solution routines.New version program summaryProgram title: PSsolverCatalogue identifier: ADPR_v2_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADPR_v2_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 2302No. of bytes in distributed program, including test data, etc.: 31962Distribution format: tar.gzProgramming language: Maple 14 (also tested using Maple 15 and 16).Computer: Intel Pentium Processor P6000, 1.86 GHz.Operating system: Windows 7.RAM: 4 GB DDR3 MemoryClassification: 4.3.Catalogue identifier of previous version: ADPR_v1_0Journal reference of previous version: Comput. Phys. Comm. 144 (2002) 46Does the new version supersede the previous version?: YesNature of problem: Symbolic solution of first order differential equations via the Prelle–Singer method.Solution method: The method of solution is based on the standard Prelle–Singer method, with extensions for the cases when the FOODE contains elementary functions. Additionally, an extension of our own which solves FOODEs with Liouvillian solutions is included.Reasons for new version: The program was not running anymore due to changes in the latest versions of Maple. Additionally, we corrected/changed some bugs/details that were hampering the smoother functioning of the routines.Summary of revisions:• As time went by, many commands in Maple were deprecated. So, in order to make the program able to run with the newer versions, we have checked and changed some of those. For instance, the command sum had changed, and some program lines were substituted so that the package works properly.• In the old version we must supply the degree of the Darboux polynomials we want to determine. In the present version the user can set the degree by typing Deg = number in the command call (e.g., PSsolve(ode, Deg =3); telling the command PSsolve that it must use Darboux polynomials of degree up to three). If the user does not specify the degree, the routines use, as default, the degree 1.Restrictions: If the integrating factor for the FOODE under consideration has factors of high degree in the dependent and independent variables and in the elementary functions appearing in the FOODE, the package may spend a long time finding the solution. Also, when dealing with FOODEs containing elementary functions, it is essential that the algebraic dependency between them is recognized. If that does not happen, our program can miss some solutions.Unusual features: Our implementation of the Prelle–Singer approach not only solves FOODEs, but can also be used as a research tool that allows the user to follow all the steps of the procedure. For example, the Darboux polynomials (eigenpolynomials) of the D-operator associated with a FOODE (see Section 4) can be calculated. In addition, our package is successful in solving FOODEs that were not solved by some of the most commonly available solvers. Finally, our package implements a theoretical extension (for details, see [1,2]) to the original Prelle–Singer approach that enhances its scope, allowing it to tackle some FOODEs whose solutions involve non-elementary Liouvillian functions.Running time: This depends strongly on the FOODE, but usually under 2 seconds when running our ‘arena’ test file: The non linear FOODEs presented in the book by Kamke [3]. These times were obtained using an Intel Pentium Processor P6000, 1.86 GHz, with 4 GB RAM.