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Solving First Order Autonomous Algebraic Ordinary Differential Equations by Places

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Abstract

Given a first order autonomous algebraic ordinary differential equation, we present a method for computing formal power series solutions by means of places. We provide an algorithm for computing a full characterization of possible initial values, classified in terms of the number of distinct formal power series solutions extending them. In addition, if a particular initial value is given, we present a second algorithm that computes all the formal power series solutions, up to a suitable degree, corresponding to it. Furthermore, when the ground field is the field of the complex numbers, we prove that the computed formal power series solutions are all convergent in suitable neighborhoods.

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Acknowledgements

The authors are 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 first author is also supported by the strategic program “Innovatives OÖ 2020” by the Upper Austrian Government and by the Austrian Science Fund (FWF): P 31327-N32.

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Authors and Affiliations

  1. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Linz, Austria

    Sebastian Falkensteiner

  2. Research Group ASYNACS (Ref.CT-CE2019/683). Dpto. de Física y Matemáticas, Universidad de Alcalá, Madrid, Spain

    J. Rafael Sendra

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  1. Sebastian Falkensteiner
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  2. J. Rafael Sendra
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Correspondence to Sebastian Falkensteiner.

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Falkensteiner, S., Sendra, J.R. Solving First Order Autonomous Algebraic Ordinary Differential Equations by Places. Math.Comput.Sci. 14, 327–337 (2020). https://doi.org/10.1007/s11786-019-00431-6

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  • DOI: https://doi.org/10.1007/s11786-019-00431-6

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