Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1

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$$\begin{array}{} \displaystyle yy'=(\alpha x^{m+k-1}+\beta x^{m-k-1})y+\gamma x^{2m-2k-1}, \quad y'=\frac{dy}{dx} \end{array}$$where a, b, c ∈ ℂ, m, k ∈ ℤ and$$\begin{array}{} \displaystyle \alpha=a(2m+k) \quad \beta=b(2m-k), \quad \gamma=-(a^2mx^{4k}+cx^{2k}+b^2m). \end{array}$$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.