Darboux integrating factors: Inverse problems

Colin Christopher, Jaume Llibre, Chara Pantazi, Sebastian Walcher

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We discuss planar polynomial vector fields with prescribed Darboux integrating factors, in a nondegenerate affine geometric setting. We establish a reduction principle which transfers the problem to polynomial solutions of certain meromorphic linear systems, and show that the space of vector fields with a given integrating factor, modulo a subspace of explicitly known "standard" vector fields, has finite dimension. For several classes of examples we determine this space explicitly. © 2010 Elsevier Inc.
Original languageEnglish
Pages (from-to)1-25
JournalJournal of Differential Equations
Issue number1
Publication statusPublished - 1 Jan 2011


  • Integrating factor
  • Invariant algebraic curve
  • Polynomial differential system

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    Christopher, C., Llibre, J., Pantazi, C., & Walcher, S. (2011). Darboux integrating factors: Inverse problems. Journal of Differential Equations, 250(1), 1-25. https://doi.org/10.1016/j.jde.2010.10.013