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Algebraic General Solutions of First Order Algebraic ODEs

  • Ngoc Thieu Vo
  • Franz Winkler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9301)

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.

Keywords

General Solution Total Degree Symbolic Computation Minimal Polynomial Irreducible Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    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)Google Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    Carnicer, M.M.: The Poincar Problem in the Nondicritical Case. Ann. of Math. 140(2), 289294 (1994)Google Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    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)Google Scholar
  6. 6.
    Kolchin, E.R.: Differential Algebra and Algebraic Groups. Pure and Applied Mathematics, vol. 54. Academic Press, New York (1973)Google Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    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)Google Scholar
  9. 9.
    Ritt, J.F.: Differential Algebra. Dover Publications Inc., New York (1955)Google Scholar
  10. 10.
    Schinzel, A.: Polynomials with Special Regard to Reducibility. Cambridge University Press (2000)Google Scholar
  11. 11.
    Sendra, J.R., Winkler, F., Pérez-Díaz, S.: Rational Algebraic Curves, A Computer Algebra Approach. Algorithms and Computation in Mathematics, vol. 22. Springer-Verlag, Heidelberg (2008)Google Scholar
  12. 12.
    Singer, M.F.: Liouvillian first integrals of differential equations. Transaction of the American Mathematics Society 333(2), 673–688 (1992)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Research Institute for Symbolic Computation (RISC)Johannes Kepler UniversityLinzAustria

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