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.
