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We study differential equations where is a formal series in with coefficients in some field of generalized power series with finite rank . Our purpose is to express the support , i.e. the set of exponents, of the elements that are solutions, in terms of the supports of the coefficients of the equation, namely .
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 equationthe idea of our proof is to interpret a formal power series solution...
Let be analytic functions in the unit disk . For the authors consider the differential subordination and the differential equation of the Briot-Bouquet type:
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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