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Rational general solutions of higher order algebraic odes

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Abstract

This paper generalizes the method of Ngô and Winkler (2010, 2011) for finding rational general solutions of a first order non-autonomous algebraic ordinary differential equation (AODE) to the case of a higher order AODE, provided a proper parametrization of its solution hypersurface. The authors reduce the problem of finding the rational general solution of a higher order AODE to finding the rational general solution of an associated system. The rational general solutions of the original AODE and its associated system are in computable 1-1 correspondence. The authors give necessary and sufficient conditions for the associated system to have a rational solution based on proper reparametrization of invariant algebraic space curves. The authors also relate invariant space curves to first integrals and characterize rationally solvable systems by rational first integrals.

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Correspondence to Yanli Huang.

Additional information

This research was supported by the Austrian Science Foundation (FWF) via the Doctoral Program “Computational Mathematics” under Grant No. W1214, Project DK11, the Project DIFFOP under Grant No. P20336-N18, the SKLSDE Open Fund SKLSDE-2011KF-02, the National Natural Science Foundation of China under Grant No. 61173032, the Natural Science Foundation of Beijing under Grant No. 1102026, and the China Scholarship Council.

This paper was recommended for publication by Editor LI Ziming.

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Huang, Y., Ngô, L.X.C. & Winkler, F. Rational general solutions of higher order algebraic odes. J Syst Sci Complex 26, 261–280 (2013). https://doi.org/10.1007/s11424-013-1252-0

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  • DOI: https://doi.org/10.1007/s11424-013-1252-0

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