Bulletin of Mathematical Sciences

, Volume 3, Issue 1, pp 19–71 | Cite as

Loewy decomposition of linear differential equations

  • Fritz SchwarzEmail author
Open Access


This paper explains the developments on factorization and decomposition of linear differential equations in the last two decades. The results are applied for developing solution procedures for these differential equations. Although the subject is more than 100 years old, it has been rediscovered as recently as about 20 years ago. A fundamental ingredient has been the easy availability of symbolic computation systems to accomplish the extensive calculations usually involved in applications; to this end the interactive website has been provided. Although originally only developed for ordinary equations, it has been extended to large classes of partial equations as well. In the first part Loewy’s results for ordinary equations are outlined. Thereafter those results of differential algebra are summarized that are required for extending Loewy’s theory to partial equations. In the remaining part a fairly complete discussion of second- and some third-order partial differential equations in the plane is given; it is shown that Loewy’s result remains essentially true for these equations. Finally, several open problems and possible extensions are discussed.


Linear differential equations Factorization Loewy decomposition 

Mathematics Subject Classification (2000)

Primary 54C40 14E20 Secondary 46E25 20C20 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


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© The Author(s) 2012

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Fraunhofer GesellschaftInstitut SCAISankt AugustinGermany

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