Liouvillian first integrals of differential equations
Author:
Michael F. Singer
Journal:
Trans. Amer. Math. Soc. 333 (1992), 673688
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 , has a solution satisfying a liouvillian relation then either and are algebraically dependent or the system has a liouvillian first integral of the form where and and rational functions of and . 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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 J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput. 2 (1986). MR 839134 (88c:12011)
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 S. Lang, Algebra, AddisonWesley, 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)
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 M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc. 279 (1983). MR 704611 (85d:12008)
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 R. H. Risch, The problem of integration in finite terms, Trans. Amer. Math. Soc. 139 (1969). MR 0237477 (38:5759)
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 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
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DOI:
http://dx.doi.org/10.1090/S0002994719921062869X
PII:
S 00029947(1992)1062869X
Article copyright:
© Copyright 1992
American Mathematical Society
