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A differential Puiseux theorem in generalized series fields of finite rank

Mickaël Matusinski (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

We study differential equations F ( y , ... , y ( n ) ) = 0 where F is a formal series in y , y , ... , y ( n ) with coefficients in some field of generalized power series 𝕂 r with finite rank r * . Our purpose is to express the support Supp y 0 , i.e. the set of exponents, of the elements y 0 𝕂 r that are solutions, in terms of the supports of the coefficients of the equation, namely Supp F .

A new proof of multisummability of formal solutions of non linear meromorphic differential equations

Jean-Pierre Ramis, Yasutaka Sibuya (1994)

Annales de l'institut Fourier

We give a new proof of multisummability of formal power series solutions of a non linear meromorphic differential equation. We use the recent Malgrange-Ramis definition of multisummability. The first proof of the main result is due to B. Braaksma. Our method of proof is very different: Braaksma used Écalle definition of multisummability and Laplace transform. Starting from a preliminary normal form of the differential equation x d y d x = G 0 ( x ) + λ ( x ) + A 0 y + x μ G ( x , y ) , the idea of our proof is to interpret a formal power series solution...

A note on Briot-Bouquet-Bernoulli differential subordination

Stanisława Kanas, Joanna Kowalczyk (2005)

Commentationes Mathematicae Universitatis Carolinae

Let p , q be analytic functions in the unit disk 𝒰 . For α [ 0 , 1 ) the authors consider the differential subordination and the differential equation of the Briot-Bouquet type: p 1 - α ( z ) + z p ' ( z ) δ p α ( z ) + λ p ( z ) h ( z ) , z 𝒰 , q 1 - α...

A simple method for constructing non-liouvillian first integrals of autonomous planar systems

Axel Schulze-Halberg (2006)

Czechoslovak Mathematical Journal

We show that a transformation method relating planar first-order differential systems to second order equations is an effective tool for finding non-liouvillian first integrals. We obtain explicit first integrals for a subclass of Kukles systems, including fourth and fifth order systems, and for generalized Liénard-type systems.

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