Steen's equation and its generalizations
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.
scientific article; zbMATH DE number 1346973