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Transactions of the American Mathematical Society

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Liouvillian first integrals of differential equations

Author: Michael F. Singer
Journal: Trans. Amer. Math. Soc. 333 (1992), 673-688
MSC: Primary 12H05; Secondary 34A99
MathSciNet review: 1062869
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Abstract: Liouvillian functions are functions that are built up from rational functions using exponentiation, integration, and algebraic functions. We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liouvillian function of several variables vanishing on the curve defined by this solution, then the system has a liouvillian first integral, that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set. We can refine this result in special cases to show that the first integral must be of a very special form. For example, we can show that if the system $ dx/dz = P(x,y)$, $ dy/dz = Q(x,y)$ has a solution $ (x(z),y(z))$ satisfying a liouvillian relation then either $ x(z)$ and $ y(z)$ are algebraically dependent or the system has a liouvillian first integral of the form $ F(x,y) = \smallint RQ\,dx - RP\,dy$ where $ R = \exp (\smallint U\,dx + V\,dy)$ and $ U$ and $ V$ rational functions of $ x$ and $ y$ . We can also reprove an old result of Ritt stating that a second order linear differential equation has a nonconstant solution satisfying a liouvillian relation if and only if all of its solutions are liouvillian.

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  • [BODI77] W. E. Boyce and R. C. DiPrima, Elementary differential equations, Third ed., Wiley, New York, 1977.
  • [INCE56] E. L. Ince, Ordinary differential equations, Dover, New York, 1956. MR 0010757 (6:65f)
  • [JOU79] J. P. Jouanalou, Equations de Pfaff algebriques, Lecture Notes in Math., vol. 708, Springer-Verlag, Berlin and New York, 1979. MR 537038 (81k:14008)
  • [KAP57] I. Kaplansky, An introduction to differential algebra, Hermann, Paris, 1957. MR 0093654 (20:177)
  • [KOL73] E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, New York, 1973. MR 0568864 (58:27929)
  • [KOV86] J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. 2 (1986). MR 839134 (88c:12011)
  • [KRA82] S. G. Krantz, Function theory of several complex variables, Wiley, New York, 1982. MR 635928 (84c:32001)
  • [LANG65] S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1965. MR 0197234 (33:5416)
  • [POOLE60] E. G. C. Poole, Introduction to the theory of linear differential equations, Dover, New York, 1960. MR 0111886 (22:2746)
  • [PRSI83] M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983). MR 704611 (85d:12008)
  • [RISCH69] R. H. Risch, The problem of integration in finite terms, Trans. Amer. Math. Soc. 139 (1969). MR 0237477 (38:5759)
  • [RITT27] J. F. Ritt, On the integration in finite terms of linear differential equations of the second order, Bull Amer. Math. Soc. 33 (1927). MR 1561321
  • [RITT48] -, Integration in finite terms, Columbia Univ. Press, New York, 1948.
  • [ROS69] M. Rosenlicht, On the explicit solvability of certain transcendental equations, Inst. Hautes, Études Sci. Publ. Math. 36 (1969). MR 0258808 (41:3454)
  • [ROS76] -, On Liouville's theory of elementary functions, Pacific J. Math. 65 (1976).
  • [SEI58] A. Seidenberg, Abstract differential algebra and the analytic case, Proc. Amer. Math. Soc. 9 (1958). MR 0093655 (20:178)
  • [SEI69] -, Abstract differential algebra and the analytic case. II, Proc. Amer. Math. Soc. 23 (1969). MR 0248122 (40:1376)
  • [SIN76] M. F. Singer, Solutions of linear differential equations in function fields of one variable, Proc. Amer. Math. Soc. 54 (1976). MR 0387260 (52:8103)

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