Algebraic differential equations of the first order free from parametric singularities from the differential-algebraic standpoint

Michihiko MATSUDA

doi:10.2969/jmsj/03030447

Michihiko MATSUDA

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JOURNAL FREE ACCESS

1978 Volume 30 Issue 3 Pages 447-455

DOI https://doi.org/10.2969/jmsj/03030447

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Published: 1978 Received: October 18, 1976 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: ABSTRACT Details: Right : A differential-algebraic definition for an algebraic differential equation of the first order to be free from parametric singularities will be given. From this standpoint we shall prove three theorems which are essentially due to Briot and Bouquet, Fuchs, and Poincaré respectively.

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Right : [1] A. R. Forsyth, Theory of differential equations, Part II, Ordinary equations, not linear, II Cambridge Univ. Press, London, 1906.
[2] L. Fuchs, Über Differentialgleichungen, deren Integrale feste Verzweigungspunkte besitzen, S.-B. der Königl. preussischen Akad. Wiss. Berlin, 32 (1884), 699-710.
[3] E. R. Kolchin, Galois theory of differential fields, Amer. J. Math., 75 (1953), 753-824.
[4] P. Painlevé, Mémoire sur les équations différentielles du premier ordre, Ann. Sci. Ecole Norm. Sup., 3^e Série 8 (1891), 9-58, 103-140, 201-226, 267-284; 9 (1892), 9-30, 101-144, 283-308.
[5] P. Painlevé, Leçons sur la théorie analytique des équations différentielles, professées à Stockholm (1895), Hermann, Paris, 1897; OEuvres, Tome I, 205-798, Centre National de la Recherche Scientifique, Paris, 1972.
[6] É. Picard, Traité d'analyse, Tome III, 2^e Édition, Gauthier-Villars, Paris, 1908.
[7] H. Poincaré, Sur un théorème de M. Fuchs, Acta Math., 7 (1885), 1-32.
[8] M. Rosenlicht, On the explicit solvability of certain transcendental equations, Inst. Hautes Études Sci. Publ. Math., no. 36 (1969), 15-22.
[9] M. Rosenlicht, An analogue of L'Hospital's rule, Proc. Amer. Math. Soc., 37 (1973), 369-373.
[10] H. Weber, Lehrbuch der Algebra, Band III, 2. Aufl., Druck und Verlag von Friedr, Vieweg & Sohn Akt.-Ges., Braunschweig, 1908.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

Abstract

A differential-algebraic definition for an algebraic differential equation of the first order to be free from parametric singularities will be given. From this standpoint we shall prove three theorems which are essentially due to Briot and Bouquet, Fuchs, and Poincaré respectively.

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