I have added some new material, most of it on the period from the mid-1880s (the
notional terminus of the first edition) to the early 1900s. The Riemann-Hilbert
problem, as it is called, which concerns the existence of linear ordinary differential
equations whose solutions have prescribed behaviour at the singular points, was
taken to be solved when I wrote the first edition. Affirmative answers were attributed
to Plemelj [1908] and Birkhoff [1913], and these seemed to lie outside the period
under consideration. Then around 1990 it emerged that there was a significant gap
in the proofs, and Anosov and Bolibruch showed that in fact the Riemann-Hilbert
problem cannot be solved in general. I have therefore added a few pages on the
history of this topic; the details of the achievement of Anosov and Bolibruch can be
found in their book [1994]. Another account of the modern state-of-the-art can be
found in the paper by Varadarajan [1991], in which the wider class of meromorphic
differential equations is also considered.
opened the tradition of Picard-Vessiot theory, otherwise