## Elementary first integrals of differential equations

HTML articles powered by AMS MathViewer

- by M. J. Prelle and M. F. Singer
- Trans. Amer. Math. Soc.
**279**(1983), 215-229 - DOI: https://doi.org/10.1090/S0002-9947-1983-0704611-X
- PDF | Request permission

## Abstract:

We show that if a system of differential equations has an elementary first integral (i.e. a first integral expressible in terms of exponentials, logarithms and algebraic functions) then it must have a first integral of a very simple form. This unifies and extends results of Mordukhai-Boltovski, Ritt and others and leads to a partial algorithm for finding such integrals.## References

- J. P. Jouanolou,
*Équations de Pfaff algébriques*, Lecture Notes in Mathematics, vol. 708, Springer, Berlin, 1979 (French). MR**537038**
C. Mack, - Joel Moses and Richard Zippel,
*An extension of Liouville’s theorem*, Symbolic and algebraic computation (EUROSAM ’79, Internat. Sympos., Marseille, 1979) Lecture Notes in Comput. Sci., vol. 72, Springer, Berlin-New York, 1979, pp. 426–430. MR**575703** - 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 - Robert H. Risch,
*The solution of the problem of integration in finite terms*, Bull. Amer. Math. Soc.**76**(1970), 605–608. MR**269635**, DOI 10.1090/S0002-9904-1970-12454-5 - J. F. Ritt,
*On the integrals of elementary functions*, Trans. Amer. Math. Soc.**25**(1923), no. 2, 211–222. MR**1501240**, DOI 10.1090/S0002-9947-1923-1501240-7
—, - Maxwell Rosenlicht,
*On Liouville’s theory of elementary functions*, Pacific J. Math.**65**(1976), no. 2, 485–492. MR**447199** - Maxwell Rosenlicht and Michael Singer,
*On elementary, generalized elementary, and Liouvillian extension fields*, Contributions to algebra (collection of papers dedicated to Ellis Kolchin), Academic Press, New York, 1977, pp. 329–342. MR**0466093** - Michael F. Singer,
*Elementary solutions of differential equations*, Pacific J. Math.**59**(1975), no. 2, 535–547. MR**389874** - Michael F. Singer,
*Functions satisfying elementary relations*, Trans. Amer. Math. Soc.**227**(1977), 185–206. MR**568865**, DOI 10.1090/S0002-9947-1977-0568865-2

*Integration of affine forms over elementary functions*, Computer Science Department, University of Utah, Technical Report, VCP-39, 1976. D. Mordukhai-Boltovski,

*Researches on the integration in finite terms of differential equations of the first order*, Communications de la Société Mathématique de Kharkov,

**X**(1906-1909), pp. 34-64, 231-269. (Russian) Translation of pp. 34-64, B. Korenblum and M. J. Prelle, SIGSAM Bulletin, Vol. 15, No. 2, May 1981, pp. 20-32.

*Oeuvres de Paul Painlevé*, Vols. I and II, Editions CNRS, Paris, 1972.

*Oeuvres de Henri Pointuré*, Vol. III, Gauthier-Villars, Paris, 1934. M. J. Prelle, Thesis, Rensselaer Polytechnic Institute, 1982.

*Integration in finite terms. Liouville’s theory of elementary methods*, Columbia Univ. Press, New York, 1948.

## Bibliographic Information

- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**279**(1983), 215-229 - MSC: Primary 12H05
- DOI: https://doi.org/10.1090/S0002-9947-1983-0704611-X
- MathSciNet review: 704611