Liouvillian first integrals of differential equations
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- by Michael F. Singer PDF
- Trans. Amer. Math. Soc. 333 (1992), 673-688 Request permission
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.References
-
W. E. Boyce and R. C. DiPrima, Elementary differential equations, Third ed., Wiley, New York, 1977.
- E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757
- J. P. Jouanolou, Équations de Pfaff algébriques, Lecture Notes in Mathematics, vol. 708, Springer, Berlin, 1979 (French). MR 537038
- Irving Kaplansky, An introduction to differential algebra, Publ. Inst. Math. Univ. Nancago, No. 5, Hermann, Paris, 1957. MR 0093654
- E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York-London, 1973. MR 0568864
- Jerald J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. 2 (1986), no. 1, 3–43. MR 839134, DOI 10.1016/S0747-7171(86)80010-4
- Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
- Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
- E. G. C. Poole, Introduction to the theory of linear differential equations, Dover Publications, Inc., New York, 1960. MR 0111886
- M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983), no. 1, 215–229. MR 704611, DOI 10.1090/S0002-9947-1983-0704611-X
- Robert H. Risch, The problem of integration in finite terms, Trans. Amer. Math. Soc. 139 (1969), 167–189. MR 237477, DOI 10.1090/S0002-9947-1969-0237477-8
- J. F. Ritt, On the integration in finite terms of linear differential equations of the second order, Bull. Amer. Math. Soc. 33 (1927), no. 1, 51–57. MR 1561321, DOI 10.1090/S0002-9904-1927-04307-5 —, Integration in finite terms, Columbia Univ. Press, New York, 1948.
- Maxwell Rosenlicht, On the explicit solvability of certain transcendental equations, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 15–22. MR 258808 —, On Liouville’s theory of elementary functions, Pacific J. Math. 65 (1976).
- A. Seidenberg, Abstract differential algebra and the analytic case, Proc. Amer. Math. Soc. 9 (1958), 159–164. MR 93655, DOI 10.1090/S0002-9939-1958-0093655-0
- A. Seidenberg, Abstract differential algebra and the analytic case. II, Proc. Amer. Math. Soc. 23 (1969), 689–691. MR 248122, DOI 10.1090/S0002-9939-1969-0248122-5
- Michael F. Singer, Solutions of linear differential equations in function fields of one variable, Proc. Amer. Math. Soc. 54 (1976), 69–72. MR 387260, DOI 10.1090/S0002-9939-1976-0387260-7
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 333 (1992), 673-688
- MSC: Primary 12H05; Secondary 34A99
- DOI: https://doi.org/10.1090/S0002-9947-1992-1062869-X
- MathSciNet review: 1062869