Loewy decomposition of linear differential equations

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 http://www.alltypes.de 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.