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Acta Applicandae Mathematicae

, Volume 120, Issue 1, pp 101–126 | Cite as

Inverse Problems in Darboux’ Theory of Integrability

  • Colin Christopher
  • Jaume Llibre
  • Chara Pantazi
  • Sebastian WalcherEmail author
Article

Abstract

The Darboux theory of integrability for planar polynomial differential equations is a classical field, with connections to Lie symmetries, differential algebra and other areas of mathematics. In the present paper we introduce the concepts, problems and inverse problems, and we outline some recent results on inverse problems. We also prove a new result, viz. a general finiteness theorem for the case of prescribed integrating factors. A number of relevant examples and applications is included.

Keywords

Planar polynomial vector field Invariant curve Elementary integrability Integrating factor 

Notes

Acknowledgements

J. Llibre and C. Pantazi are partially supported by the MICIIN/FEDER grant MTM2008-03437. JL is additionally partially supported by a CIRIT grant number 2009SGR410 and by ICREA Academia. CP is additionally partial supported by the MICIIN/FEDER grant number MTM2009-06973 and by the Generalitat de Catalunya grant number 2009SGR859. S. Walcher acknowledges the hospitality and support of the CRM and the Mathematics Department at UAB during visits when this manuscript was prepared.

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Colin Christopher
    • 1
  • Jaume Llibre
    • 2
  • Chara Pantazi
    • 3
  • Sebastian Walcher
    • 4
    Email author
  1. 1.Department of Mathematics and StatisticsUniversity of PlymouthPlymouthUK
  2. 2.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain
  3. 3.Departament de Matemàtica Aplicada IUniversitat Politècnica de Catalunya, (EPSEB)BarcelonaSpain
  4. 4.Lehrstuhl A für MathematikRWTH AachenAachenGermany

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