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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.
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.
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