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Puiseux Series and Algebraic Solutions of First Order Autonomous AODEs – A MAPLE Package

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Maple in Mathematics Education and Research (MC 2020)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 1414))

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Abstract

There exist several methods for computing exact solutions of algebraic differential equations. Most of the methods, however, do not ensure existence and uniqueness of the solutions and might fail after several steps, or are restricted to linear equations. The authors have presented in previous works a method to overcome this problem for autonomous first order algebraic ordinary differential equations and formal Puiseux series solutions and algebraic solutions. In the first case, all solutions can uniquely be represented by a sufficiently large truncation and in the latter case by its minimal polynomial.

The main contribution of this paper is the implementation, in a MAPLE package named FirstOrderSolve, of the algorithmic ideas presented therein. More precisely, all formal Puiseux series and algebraic solutions, including the generic and singular solutions, are computed and described uniquely. The computation strategy is to reduce the given differential equation to a simpler one by using local parametrizations and the already known degree bounds.

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Acknowledgements

The first author was supported by the bilateral project ANR-17-CE40-0036 and DFG-391322026 SYMBIONT. The second author was partially supported by Agencia Estatal de Investigación PID2019-105621GB-I00. The third and fourth authors were partially supported by FEDER/Ministerio de Ciencia, Innovación y Universidades Agencia Estatal de Investigación/MTM2017-88796-P (Symbolic Computation: new challenges in Algebra and Geometry together with its applications). The third author was also supported by the Austrian Science Fund (FWF): P 31327-N32. The fourth author is member of the Research Group ASYNACS (Ref. CT-CE2019/683).

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Correspondence to Sebastian Falkensteiner .

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Boulier, F., Cano, J., Falkensteiner, S., Sendra, J.R. (2021). Puiseux Series and Algebraic Solutions of First Order Autonomous AODEs – A MAPLE Package. In: Corless, R.M., Gerhard, J., Kotsireas, I.S. (eds) Maple in Mathematics Education and Research. MC 2020. Communications in Computer and Information Science, vol 1414. Springer, Cham. https://doi.org/10.1007/978-3-030-81698-8_7

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  • DOI: https://doi.org/10.1007/978-3-030-81698-8_7

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