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An application of Ritt’s low power theorem

Published online by Cambridge University Press:  22 January 2016

Michihiko Matsuda*
Affiliation:
Department of Mathematics Osaka University
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Abstract

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Consider an algebraic differential equation F = 0 of the first order. A rigorous definition will be given to the classical concept of “particular solutions” of F = 0. By Ritt’s low power theorem we shall prove that a singular solution of F = 0 belongs to the general solution of F if and only if it is a particular solution of F = 0.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1977

References

[1] Forsyth, A. R., Theory of differential equations, Part II, Ordinary equations, not linear, Vol. II, Cambridge Univ. Press, London, 1906.Google Scholar
[2] Iwasawa, K., Theory of algebraic function fields (in Japanese), Iwanami Shoten, Tokyo, 1952.Google Scholar
[3] Kolchin, E. R., Galois theory of differential fields, Amer. J. Math., 75 (1953), 753824.CrossRefGoogle Scholar
[4] Picard, E., Traite d’analyse, Tome II, 2 e Edition, Gauthier-Villars, Paris, 1905.Google Scholar
[5] Ritt, J. F., Differential algebra, Amer. Math. Soc. Colloq. Publ. Vol. 33, New York, 1950.Google Scholar