Global theory of ordinary linear homogeneous differential equations in the real domain. I
type: Item
instance of: scholarly article
title: Global theory of ordinary linear homogeneous differential equations in the real domain. I
zbMATH Open document ID: 0643.34011
DOI: 10.1007/BF01836159
author: František Neuman
published in: Aequationes Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137209
review text: In this long paper the author carries out an in-depth investigation and presents a survey of the global properties of linear homogeneous differential equations of the nth order, \(n\geq 2\), with real continuous coefficients defined on open intervals of reals. The set of these equations will be denoted by \({\mathbb{A}}_ n\). Each equation of the set \({\mathbb{A}}_ n\) can be written in the form \[ P_ n(x,y,I)\quad \equiv \quad y^{(n)}+P_{n-1}(x)y^{(n-1)}+...+P_ 0(x)y=0, \] where \(P_ i\in C^ 0(I)\) for \(i=0,1,...,n-1\), the interval \(I\subset {\mathbb{R}}\) is open and \(n\geq 2\) is an integer. The author compares his results with previous local investigations and succeeds in obtaining new results describing global properties of linear homogeneous differential equations of an arbitrary order n, \(n\geq 2\), e.g., he extends Boriuokla's criterion for the global equivalence of the second order linear differential equations into a new criterion of global equivalence of equations of an arbitrary order n from \({\mathbb{A}}_ n\), \(n\geq 3\). In the last section of the paper the author introduces stationary groups, i.e. the groups formed by all global transformations that transform a given ordinary linear differential equation into itself which describe completely the structure of all global transformations in each class of globally equivalent equations.
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 65Lxx
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 4048102
zbMATH Keywords: iterative equation
zbMATH Keywords: effective equations
zbMATH Keywords: linear homogeneous differential equations
zbMATH Keywords: Boriuokla's criterion
zbMATH Keywords: second order linear differential equations
zbMATH Keywords: global equivalence
zbMATH Keywords: stationary groups
reviewed by: F. M. Ragab
MaRDI profile type: Publication
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OpenAlex ID: W4247924997
description: scientific article; zbMATH DE number 4048102