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

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′=(αxm+k−1+βxm−k−1)y+γx2m−2k−1,y′=dydx where a, b, c ∈ ℂ, m, k ∈ ℤ and α=a(2m+k)β=b(2m−k),γ=−(a2mx4k+cx2k+b2m). This family is very important because...