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The aim of this paper is the analysis, from algebraic point of view andsingularities studies, of the 5-parametric family of differential equations  \begin{equation*}\label{folpz}  yy'=(\alpha x^{m+k-1}+\beta x^{m-k-1})y+\gamma x^{2m-2k-1}, \quady'=\frac{dy}{dx}  \end{equation*}  where $a,b,c\in \mathbb{C}$, $m,k\in \mathbb{Z}$ and  $$\alpha=a(2m+k) \quad \beta=b(2m-k), \quad\gamma=-(a^2mx^{4k}+cx^{2k}+b^2m).$$ This family is very important becauseinclude Van Der Pol equation. Moreover, this family seems to appear as exercisein the celebrated book of Polyanin and Zaitsev. Unfortunately, the exercisepresented a typo which does not allow to solve correctly it. We present thecorrected exercise, which corresponds to the title of this paper. We solve theexercise and afterwards we make algebraic and of singularities studies to thisfamily of differential equations. To illustrate the qualitative and algebraictechniques we present an example of a biparametric quadratic Polyanin-Zaitsevvector field.

Algebraic and qualitative remarks about the family $yy'= (\alpha x^{m+k-1} + \beta x^{m-k-1})y + \gamma x^{2m-2k-1}$