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.
The analysis of dynamical systems has been a topic of great interest for researches mathematical sciences for a long times. The implementation of several devices and tools have been useful in the finding of solutions as well to describe common behaviors of parametric families of these systems. In this paper we study deeply a particular parametric family of differential equations, the so-called Linear Polyanin-Zaitsev Vector Field, which has been introduced in a general case in [1] as a correction of a family presented in [2]. Linear Polyanin-Zaitsev Vector Field is transformed into a Liénard equation and in particular we obtain the Van Der Pol equation. We present some algebraic and qualitative results to illustrate some interactions between algebra and the qualitative theory of differential equations in this parametric family.
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