Inverse Approach In The Study Of Ordinary Differential Equations

2009/02/26 by Rafael Ramírez, Ramirez, Rafael, Natalia Sadovskaia +1
Mathematics · #14P25 #34A34 #34C05 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.0902.4649

openalex publication_date 2009/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We extend the Eruguin result exposed in the paper "Construction of the whole set of ordinary differential equations with a given integral curve" published in 1952 and construct a differential system in ℝ^N which admits a given set of the partial integrals, in particular we study the case when theses functions are polynomials. We construct a non-Darboux integrable planar polynomial system of degree n with one invariant irreducible algebraic curve g(x,y)=0. For this system we analyze the Darboux integrability, Poincare's problem and 16th's Hilbert problem for algebraic limit cycles.

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• Partial integrals of ordinary differential systems
• A family of quadratic polynomial differential systems with algebraic solutions of arbitrary high degree
• Rational curves and ordinary differential equations
• The differential equation satisfied by a plane curve of degree n