Integrability of planar polynomial differential systems through linear differential equations
In this work we consider rational ordinary differential equations dy/dx = Q(x, y)/P(x, y), with Q(x, y) and P(x, y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-...
| Autores: | , , |
|---|---|
| Tipo de recurso: | artículo |
| Estado: | Versión enviada para evaluación y publicación |
| Fecha de publicación: | 2006 |
| País: | España |
| Institución: | Universitat de Lleida (UdL) |
| Repositorio: | Repositori Obert UdL |
| OAI Identifier: | oai:repositori.udl.cat:10459.1/57788 |
| Acceso en línea: | https://doi.org/10.1216/rmjm/1181069462 http://hdl.handle.net/10459.1/57788 |
| Access Level: | acceso abierto |
| Palabra clave: | Planar polynomial system First integral Invariant curves Darboux integrability Equacions diferencials |
| Sumario: | In this work we consider rational ordinary differential equations dy/dx = Q(x, y)/P(x, y), with Q(x, y) and P(x, y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-order homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function. We show that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral, can be constructed by using this method. We also present a new example of this kind of family. We give an analogous method for constructing rational equations but by means of a linear differential equation of first order. |
|---|