Liouvillian First Integrals of Differential Equations
Open Access
- 1 October 1992
- journal article
- Published by JSTOR in Transactions of the American Mathematical Society
- Vol. 333 (2), 673-688
- https://doi.org/10.2307/2154053
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 <!-- MATH $dx/dz = P(x,y)$ --> , <!-- MATH $dy/dz = Q(x,y)$ --> has a solution <!-- MATH $(x(z),y(z))$ --> satisfying a liouvillian relation then either and are algebraically dependent or the system has a liouvillian first integral of the form <!-- MATH $F(x,y) = \smallint RQ\,dx - RP\,dy$ --> where <!-- MATH $R = \exp (\smallint U\,dx + V\,dy)$ --> 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.
Keywords
This publication has 12 references indexed in Scilit:
- An algorithm for solving second order linear homogeneous differential equationsJournal of Symbolic Computation, 1986
- Elementary First Integrals of Differential EquationsTransactions of the American Mathematical Society, 1983
- Solutions of Linear Differential Equations in Function Fields of One VariableProceedings of the American Mathematical Society, 1976
- Abstract differential algebra and the analytic case. IIProceedings of the American Mathematical Society, 1969
- The Problem of Integration in Finite TermsTransactions of the American Mathematical Society, 1969
- On the explicit solvability of certain transcendental equationsPublications mathématiques de l'IHÉS, 1969
- An Introduction to Differential Algebra. By Irving Kaplansky. pp. 64. 800 fr. 1957. (Hermann, Paris)The Mathematical Gazette, 1959
- Abstract differential algebra and the analytic caseProceedings of the American Mathematical Society, 1958
- Introduction to the Theory of Linear Differential Equations. By E. G. C. Poole Pp. viii, 202. 17s. 6d. 1936. (Oxford)The Mathematical Gazette, 1937
- On the integration in finite terms of linear differential equations of the second orderBulletin of the American Mathematical Society, 1927