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Rational Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One

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Abstract

We give a necessary and sufficient condition for an autonomous first-order AODE to have a rational liouvillian solution. We also give an algorithm to compute a rational liouvillian general solution if it exists. The algorithm is based on the rational parametrization of the corresponding algebraic curve of the first-order autonomous AODE and the existence of a rational liouvillian element over \(\mathbb {C}\). When the corresponding algebraic curve is rational, this method covers the known cases of rational solutions and radical solutions.

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Funding

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2017.312.

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Authors and Affiliations

  1. Ho Chi Minh City University of Transport, Ho Chi Minh City, Vietnam

    Nguyen Tri Dat

  2. Quy Nhon University, Quy Nhon, Vietnam

    Ngo Lam Xuan Chau

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  1. Nguyen Tri Dat
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  2. Ngo Lam Xuan Chau
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Correspondence to Nguyen Tri Dat.

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Dat, N.T., Chau, N.L.X. Rational Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One. Acta Math Vietnam 46, 689–700 (2021). https://doi.org/10.1007/s40306-020-00404-z

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  • DOI: https://doi.org/10.1007/s40306-020-00404-z

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