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Integrability and limit cycles for Abel equations

Jean-Pierre Françoise (2011)

Banach Center Publications

Abel equations are among the most natural ordinary differential equations which have a Godbillon-Vey sequence of length 4. We show that the associated Poincaré mapping can be expressed by iterated integrals with three functions which are solutions of a system of partial differential equations.

Integrability of a linear center perturbed by a fifth degree homogeneous polynomial.

Javier Chavarriga, Jaume Giné (1997)

Publicacions Matemàtiques

In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones.

Integrability of a linear center perturbed by a fourth degree homogeneous polynomial.

Javier Chavarriga, Jaume Giné (1996)

Publicacions Matemàtiques

In this work we study the integrability of a two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We give sufficient conditions for integrability in polar coordinates. Finally we establish a conjecture about the independence of the two classes of parameters which appear in the system; if this conjecture is true the integrable cases found will be the only possible ones.

Integrable systems in the plane with center type linear part

Javier Chavarriga (1994)

Applicationes Mathematicae

We study integrability of two-dimensional autonomous systems in the plane with center type linear part. For quadratic and homogeneous cubic systems we give a simple characterization for integrable cases, and we find explicitly all first integrals for these cases. Finally, two large integrable system classes are determined in the most general nonhomogeneous cases.

Integrating factor

Jan Mařík (1994)

Mathematica Bohemica

The problem of integrating factor for ordinary differential equations is investigated. Conditions are given which guarantee that each solution of ∂ 1 F ( x , y ) + y ' ∂ 2 F ( x , y ) = 0 is also a solution of M ( x , y ) + y ' N ( x , y ) = 0 where ∂ 1 F = μ M and ∂ 2 F = μ N .

Intégration algorithmique des fonctions élémentairement transcendantes sur une courbe algébrique

J. H. Davenport (1984)

Annales de l'institut Fourier

On considère le problème de déterminer les solutions d’une équation différentielle ordinaire, dite de Risch sur une courbe algébrique. En fait une généralisation assez évidente de la méthode de Risch suffit mais elle nous permet de généraliser son algorithme d’intégration à toute extension élémentairement transcendante d’une extension algébrique des fonctions rationnelles.

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