Algebraic and qualitative remarks about the family yy'= (αx^m+k-1 + βx^m-k-1)y + γx^2m-2k-1

2018/07/10 by Primitivo B. Acosta-Humánez, Acosta-Humánez, Primitivo B., Alberto Reyes-Linero +3
Mathematics · #12H05 #34C99 #Advanced Algebra and Geometry #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1807.03551

openalex publication_date 2018/07/10 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

The aim of this paper is the analysis, from algebraic point of view and singularities studies, of the 5-parametric family of differential equations yy'=(αx^m+k-1+βx^m-k-1)y+γx^2m-2k-1, y'=(dy)/(dx) where a,b,c∈ ℂ, m,k∈ ℤ and α=a(2m+k) β=b(2m-k), γ=-(a²mx^4k+cx^2k+b²m). This family is very important because include Van Der Pol equation. Moreover, this family seems to appear as exercise in the celebrated book of Polyanin and Zaitsev. Unfortunately, the exercise presented a typo which does not allow to solve correctly it. We present the corrected exercise, which corresponds to the title of this paper. We solve the exercise and afterwards we make algebraic and of singularities studies to this family of differential equations. To illustrate the qualitative and algebraic techniques we present an example of a biparametric quadratic Polyanin-Zaitsev vector field.

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