Steen's equation and its generalizations
type: Item
instance of: scholarly article
title: Steen's equation and its generalizations
zbMATH Open document ID: 0934.34026
published in: Aequationes Mathematicae
publication date: 2000-04-10T00:00:00Z
review text: The author deals with a nonlinear differential equation investigated by \textit{A. Steen} [Forhandl. af Kjobnhaven, 1-12 (1874; JFM 06.0207.02)] in a little-known paper in 1874. Steen's equation has been rediscovered and generalized in numerous papers without any mention of the work of Steen. The author draws attention to this fact and gives a unified approach that generalizes Steen's results and leads to a notable simplification of detail. The main result is the following: Let \(I\) be an open interval on the real line, let \(p\neq 0\), \(p_1\), \(q\) be continuous complex-valued functions and let be \[ W(u,v)= uv'- u'v,\quad F(t)= \int{p_1(t)\over p(t)} dt. \] Under these assumptions the following theorem holds: Let be \(c\neq 0\) a constant and let \[ pr''+ p_1r'+ qr= {p\over e^{2F}} {c^2\over r^3},\quad e^Fr^2 s= c,\tag{ab} \] and \[ py''+ p_1y'+ qy= 0,\quad e^FW(u,v)= c.\tag{cd} \] i) If (ab) holds then \(u= r\cos s\), \(v= r\sin s\) satisfy \(u^2+ v^2\neq 0\) and (cd). ii) If \(u\), \(v\) satisfy \(u^2+ v^2\neq 0\) and (cd) then they can be written in the form \(u= r\cos s\), \(v= r\sin s\), where \(r\), \(s\) satisfy (ab). The author gives some historical remarks, discusses global existence and boundedness of the ratios \({r\over\overline r}\) and \({\overline r\over r}\) on \(I\), where \(r\) and \(\overline r\) satisfy (a) on the interval \(I\), gives a geometric representation of the solution to (c) and other properties.
reviewed by: Marko Švec
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 1346973
zbMATH Keywords: global existence
zbMATH Keywords: boundedness of ratios of two solutions
zbMATH Keywords: nonlinear differential equation
zbMATH Keywords: JFM 06.0207.02
zbMATH Keywords: Steen's equation
author: Ray Redheffer
MaRDI profile type: Publication
full work available at URL: s000100050093
OpenAlex ID: W2011878666
DOI: 10.1007/S000100050093
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description: scientific article; zbMATH DE number 1346973