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Rational Solutions of Riccati Differential Equation with Coefficients Rational


Nadhem ECHI

Universite de Tunis
El Manar, B.P. 37,
1002-TUNIS Belvedere,


This paper presents a simple and efficient method for determining the solution of Riccati differential equation with coefficients rational. In case the differential Galois group of the differential equation is reducible, we look for the rational solutions of Riccati differential equation by reducing the number of check to be made and by accelerating the search for the partial fraction decomposition of the solution. This partial fraction decomposition of solution can be used to code r. The examples demonstrate the effectiveness of the method.


  1. B. Sturmfels. Algorithms in Invariant Theory. Springer-Verlag, New York, 1993
  2. E. R. Kolchin. Algebraic matric and the Picard-Vessiot theory of homogeneous linear ordinary differential equation. Ann. of Math, 49, 1-42(1948)
  3. E. R. Kolchin. Differential Algebra and Algebraic groups, Volume 54 of Pure and Applied Math, Academic Press, New York, (1976)
  4. E. R. Kolchin. Galois theory of differential fields, Amer. J. of Math, 75, 753-824 (1953)
  5. F. Ulmer. Introduction to differential Galois theory. Cours de DEA, 2000
  6. F. Ulmer. On liouvillian solutions of linear differential equations. Appl. Algebra in Eng. Comm. and Comp., 226(2):171{193, 1992
  7. I. Gozard. Theorie de Galois, Paris (1997)
  8. I. Kaplansky. An introduction to differential algebra, Hermann, Paris , (1957)
  9. J. J. Kovacic. An Algorithm for solving second order linear homogeneous differential equations. Journal of symbolic computation, 2:3-43, 1986
  10. M. Singer and F. Ulmer. Galois groups of second and third order linear differential equations. J. Symb. Comp., 16:1{36, 1993
  11. M. Singer and F. Ulmer, Linear differential equations and products of linear forms. J. Pure and Applied Alg., 117 - 118:549-564, (1997)
  12. M. Singer and F. Ulmer. Liouvillian and algebraic solutions of second and third order linear differential equations. J. Symb. Comp., 16:37{73, (1993)
  13. M. Singer and F. Ulmer. Necessary conditions for liouvillian solutions of (third order) linear differential equations. J. of Appl. Alg. in Eng. Comm. and Comp., 6:1{22, 1995
  14. M. van der Put and F. Ulmer. Differential equations and finite groups. J. of Algebra, 226:920-966, (2000)
  15. P. A. Hendriks and M. van der Put. Galois action on solutions of a differential equation. Journal of Symbolic Computation, 19(6): 559-576, (1995)

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