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

Primitivo B. Acosta-Humánez, Alberto Reyes-Linero+1 more|arXiv (Cornell University)|Jul 10, 2018

Advanced Differential Equations and Dynamical Systems15 references3 citations

TL;DR

This paper corrects a longstanding typo in Exercise 11 of Polyanin and Zaitsev's *Handbook of Exact Solutions of Ordinary Differential Equations*, resolving the 5-parameter family of equations $yy' = (\alpha x^{m+k-1} + \beta x^{m-k-1})y + \gamma x^{2m-2k-1}$, which includes the Van der Pol equation. It provides a complete algebraic and qualitative analysis using differential Galois theory and singularity theory, deriving conditions for polynomial vector fields and classifying finite critical points, with a detailed study of a biparametric quadratic case showing stable nodes and saddle points depending on parameter signs.

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{equation*}\label{folpz} yy'=(\alpha x^{m+k-1}+\beta x^{m-k-1})y+\gamma x^{2m-2k-1}, \quad y'=\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 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.

Motivation & Objective

• To correct a persistent typo in Exercise 11 of Polyanin and Zaitsev's handbook, which invalidates the original solution.
• To provide a complete solution and algebraic analysis of the corrected 5-parameter family of ODEs, including conditions for polynomial vector fields.
• To perform qualitative analysis of the system by classifying finite critical points and their stability using singularity theory and the Poincaré compactification.
• To extend prior work on the Polyanin-Zaitsev vector field by analyzing integrability via differential Galois theory and illustrating results with a biparametric quadratic case.
• To present a complete corrigendum and solution for a system that includes the Van der Pol equation as a special case.

Proposed method

• Derive the corrected form of the ODE family using parameter substitutions and transformations, including $z = x^k$, $y = x^m(t + ax^k + bx^{-k})$.
• Transform the system into a Riccati equation and then into a second-order linear ODE using a change of variables $z = \frac{mt^2 + c_0}{ak} \frac{w'}{w}$.
• Apply differential Galois theory to assess integrability, confirming that the Riccati equation is always solvable in the sense of the theory.
• Use singularity theory and Theorem 1.1 to classify the nature of finite critical points based on the leading terms of the vector field expansion near the origin.
• Perform Poincaré compactification using charts $U_1$ and $U_2$ to analyze the behavior at infinity, particularly for the quadratic case.
• Apply the Poincaré compactification and Theorem 3.5 to classify the infinite critical point at the origin of the $U_2$ chart, determining node type based on parameter signs.

Experimental results

Research questions

• RQ1 What is the correct form of Exercise 11 in Polyanin and Zaitsev's handbook, and how does it differ from the original typo-ridden version?
• RQ2 Under what conditions on the parameters $\alpha, \beta, \gamma, m, k$ does the system yield a polynomial vector field?
• RQ3 How can the finite critical points of the system be classified, and what determines their stability (e.g., node, saddle, focus)?
• RQ4 What is the behavior of the system at infinity, and how does the Poincaré compactification reveal the structure of the global phase portrait?
• RQ5 How does the integrability of the system relate to differential Galois theory, and is the system always solvable in this sense?

Key findings

• The corrected ODE family is $yy' = (\alpha x^{m+k-1} + \beta x^{m-k-1})y + \gamma x^{2m-2k-1}$ with $\alpha = a(2m + k)$, $\beta = b(2m - k)$, $\gamma = -(a^2 m x^{4k} + c x^{2k} + b^2 m)$, and the system is integrable in the sense of differential Galois theory.
• For the biparametric quadratic case with $a=0$, $m=3/2$, $k=1/2$, the system has two finite critical points: $(0,0)$ and $(-3b^2/(2c), 0)$.
• The origin $(0,0)$ is a stable node if $b < 0$ and an unstable node if $b > 0$, based on eigenvalues $\lambda_1 = 3b/2$, $\lambda_2 = b$.
• The second critical point $(-3b^2/(2c), 0)$ is a saddle, with eigenvalues $\lambda_1 = 3b$, $\lambda_2 = -b/2$, confirming hyperbolicity.
• At infinity, the $U_2$ chart reveals a nilpotent singular point at the origin, classified as a repelling node if $c > 0$ and attracting if $c < 0$, based on parameter-dependent conditions in Theorem 3.5.
• The system has no other finite critical points when $c = 0$, and in such cases, $(0,0)$ is the only critical point, as shown in Proposition 5.3.