Generalized cofactors and nonlinear superposition principles
type: Item
instance of: scholarly article
title: Generalized cofactors and nonlinear superposition principles
zbMATH Open document ID: 1057.34002
DOI: 10.1016/S0893-9659(03)90107-8
author: Jaume Giné
author: Isaac A. García
published in: Applied Mathematics Letters
publication date: 2004-06-11T00:00:00Z
review text: In this note, the authors show that the Darboux theory of integrability can be applied to some planar polynomial vector fields \(\dot x=X(x)\) having enough invariant curves, which are not necessarily polynomial. Recall that if \(Xf=kf\) for some smooth function \(k,\) then \(f=0\) is an invariant solution of \(X\) and \(k\) is called the cofactor of \(f.\) They prove that if \(X(x)\) has degree \(m\) and if the system \(\dot x= X(x)\) possesses \(m(m+1)/2\) invariant curves \(f_i(x,y)=0,\) all of them with associated polynomial cofactors of degree \(m-1\), then \(X\) has either a first integral or an integrating factor, which can be constructed from these curves. The following illustrative example is given. The planar differential equation \(\dot x=-y+x^4,\) \(\dot y=x,\) has the invariant curves \(f_1(x,y)=2x^2{Ai}(4^{1/3}y)+{Ai}'(4^{1/3}y)=0 \) and \(f_2(x,y)=2x^2{Bi}(4^{1/3}y)+{Bi}'(4^{1/3}y)=0, \) with cofactors \(k_1(x,y)=k_2(x,y)=2x^3,\) where \({Ai}\) and \({Bi}\) are a pair of linearly independent solutions of \(w''(z)=zw(z).\) The system possesses the non-Liouvillian first integral \(H(x,y)=f_1(x,y)/f_2(x,y).\)
reviewed by: Armengol Gasull
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C14
zbMATH DE Number: 2073826
zbMATH Keywords: planar polynomial differential equations
zbMATH Keywords: transcendental solutions
zbMATH Keywords: non-Liouvillian first integral
MaRDI profile type: Publication
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full work available at URL: s0893-9659(03)90107-8
OpenAlex ID: W2049853891
description: scientific article; zbMATH DE number 2073826