Abstract
In this paper we consider the class of algebraic ordinary differential equations (AODEs), the class of planar rational systems, and discuss their algebraic general solutions. We establish for each parametrizable first order AODE a planar rational system, the associated system, such that one can compute algebraic general solutions of the one from the other and vice versa. For the class of planar rational systems, an algorithm for computing their explicit algebraic general solutions with a given rational first integral is presented. Finally an algorithm for determining an algebraic general solution of degree less than a given positive integer of parametrizable first order AODEs is proposed.
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Aroca, J.M., Cano, J., Feng, R., Gao, X.-S.: Algebraic general solutions of algebraic ordinary differential equations. In: Kauers, M. (ed.) ISSAC 2005, pp. 29–36. ACM Press, New York (2005)
Bostan, A., Chèze, G., Cluzeau, T., Weil, J.A.: Efficient algorithms for computing rational first integrals and Darboux polynomials of planar vector fields. arXiv (2013)
Carnicer, M.M.: The Poincar Problem in the Nondicritical Case. Ann. of Math. 140(2), 289294 (1994)
Feng, R., Gao, X.-S.: Rational general solution of algebraic ordinary differential equations. In: Gutierrez, J. (ed.) ISSAC 2004, pp. 155–162. ACM Press, New York (2004)
Feng, R., Gao, X.-S.: A polynomial time algorithm for finding rational general solutions of first order autonomous ODEs. Journal of Symbolic Computation 41(7), 739–762 (2006)
Kolchin, E.R.: Differential Algebra and Algebraic Groups. Pure and Applied Mathematics, vol. 54. Academic Press, New York (1973)
Ngô, L.X.C., Winkler, F.: Rational general solutions of first order non-autonomous parametrizable ODEs. Journal of Symbolic Computation 45(12), 1426–1441 (2010)
Ngô, L.X.C., Winkler, F.: Rational general solutions of planar rational systems of autonomous ODEs. Journal of Symbolic Computation 46(10), 1173–1186 (2011)
Ritt, J.F.: Differential Algebra. Dover Publications Inc., New York (1955)
Schinzel, A.: Polynomials with Special Regard to Reducibility. Cambridge University Press (2000)
Sendra, J.R., Winkler, F., Pérez-DÃaz, S.: Rational Algebraic Curves, A Computer Algebra Approach. Algorithms and Computation in Mathematics, vol. 22. Springer-Verlag, Heidelberg (2008)
Singer, M.F.: Liouvillian first integrals of differential equations. Transaction of the American Mathematics Society 333(2), 673–688 (1992)
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Vo, N.T., Winkler, F. (2015). Algebraic General Solutions of First Order Algebraic ODEs. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2015. Lecture Notes in Computer Science(), vol 9301. Springer, Cham. https://doi.org/10.1007/978-3-319-24021-3_35
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DOI: https://doi.org/10.1007/978-3-319-24021-3_35
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