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Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero

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Abstract

This paper considers the class of autonomous algebraic ordinary differential equations (AODEs) of order one, and studies their Liouvillian general solutions. In particular, let F(y, w) = 0 be a rational algebraic curve over ℂ. The authors give necessary and sufficient conditions for the autonomous first-order AODE F(y,y′) = 0 to have a Liouvillian solution over ℂ. Moreover, the authors show that a Liouvillian solution α of this equation is either an algebraic function over ℂ(x) or an algebraic function over ℂ(exp(ax)). As a byproduct, these results lead to an algorithm for determining a Liouvillian general solution of an autonomous AODE of order one of genus zero. Rational parametrizations of rational algebraic curves play an important role on this method.

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Authors and Affiliations

  1. Faculty of Basic Science, Ho Chi Minh City University of Transport, Ho Chi Minh City, Vietnam

    Tri Dat Nguyen

  2. Department of Mathematics and Statistics, Quy Nhon University, Quy Nhon City, Vietnam

    Lam Xuan Chau Ngo

Authors
  1. Tri Dat Nguyen
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  2. Lam Xuan Chau Ngo
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Corresponding author

Correspondence to Tri Dat Nguyen.

Additional information

This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.04-2017.312.

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Nguyen, T.D., Ngo, L.X.C. Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero. J Syst Sci Complex 36, 884–893 (2023). https://doi.org/10.1007/s11424-023-1246-5

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  • DOI: https://doi.org/10.1007/s11424-023-1246-5

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