Global Properties of Linear Ordinary Differential Equations

This monograph contains a description of original methods and results concern- ing global properties of linear differential equations of the nth order, n ~ 2, in the real domain. This area of research concerning second order linear differential equations was started 35 years ago by O. Boruvka. He summarized his results in the monograph "Lineare Differentialtransforrnationen 2. Ordnung", VEB, Berlin 1967 (extended version: "Linear Differential Transformations of the Second Order", The English U niv. Press, London 1971). This book deals with linear differential equations of the nth order, n ~ 2, and summarizes results in this field in a unified fashion. However, this mono- graph is by no means intended to be a survey of all results in this area. I t contains only a selection of results, which serves to illustrate the unified approach presented here. By using recent methods and results of algebra, topology, differential geometry, functional analysis, theory of functional equations and linear differential equations of the second order, and by introducing several original methods, global solutions of problems which were previously studied only locally by Kummer, Brioschi, Laguerre, Forsyth, Halphen, Lie, Stiickel and others are provided. The structure of global transformations is described by algebraic means (theory of categories: Brandt and Ehresmann groupoids), a new geometrical approach is introduced that leads to global canonical forms (in contrast to the local Laguerre-Forsyth or Halphen forms) and is suitable for the application of Cartan's moving-frame-of-reference method.