Classification of certain class of ordinary differential equations of the first order
type: Item
instance of: scholarly article
title: Classification of certain class of ordinary differential equations of the first order
zbMATH Open document ID: 1415.34066
DOI: 10.3103/S1066369X18110014
author: P. V. Bibikov
author: N. A. Safonkin
published in: Russian Mathematics
publication date: 2019-03-28T00:00:00Z
review text: In a recent work, one of the authors studied the problem of classifying second order ordinary differential equations of the form \(y'' = F(x,y)\) with \(F\) a smooth function, up to equivalence under the action of (Lie) point and contact pseudogroups [the first author, Funct. Anal. Appl. 51, No. 4, 255--262 (2017; Zbl 1402.34039); translation from Funkts. Anal. Prilozh. 51, No. 4, 16--25 (2017)]. Here, the authors tackle the same problem for equation with rational right hand side; more specifically, they consider first-order equations \[ y' = \frac{A(x) \, y + B (x)}{C(x) y+ D(x)}\eqno{(*)} \] under the action of the Cremona group \texttt{Cr(2)}. They prove that the group \(G\) of symmetries for equations in the above form consists of maps \[ g:(x,y)\to\left(\frac{\phi_1 x + \phi_2}{\psi_1 x + \psi_2}, \eta y + \xi (x) \right), \] where \[ \begin{pmatrix} \phi_1 & \phi_2 \\ \psi_1 & \psi_2 \end{pmatrix} \in \mathrm{PSL} (2 , \mathbf{C}) \ , \xi (x) \in \mathbf{C} (x)\, \eta \in \mathbf{C}. \] The Lie algebra \(\mathcal{G}\) of infinitesimal symmetries of \((*)\) consists of germs of vector fields \[ X = (\alpha x^2 + \beta x + \gamma) \partial_x + (\delta y + \sigma(x)) \partial_y ,\, \alpha,\beta,\gamma,\delta \in \mathbf{C},\sigma (x) \in \mathbf{C} (x). \] Moreover they prove that the field of differential invariants on the relevant space of infinite Jets is freely generated by two differential invariants and one invariant differentiation (all of these provided in explicit form), and give explicit necessary and sufficient conditions for the equivalence -- in the sense mentioned above -- of two equations in the considered class. A number of examples are also considered in detail.
reviewed by: Giuseppe Gaeta
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 58D19
zbMATH DE Number: 7043820
zbMATH Keywords: differential invariants
zbMATH Keywords: ordinary differential equations
zbMATH Keywords: algebraic geometry
zbMATH Keywords: Cremona group
MaRDI profile type: Publication
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full work available at URL: s1066369x18110014
OpenAlex ID: W2903360522
Wikidata QID: Q128897572
description: scientific article; zbMATH DE number 7043820