Boston, MA: Birkhäuser. xx, 338 p. $ 64.95 (2000).
This volume is a second edition of [Linear differential equations and group theory from Riemann to Poincaré (1984; Zbl 0536.01013)].
The previous edition has been reviewed very favorably. Some extensions are provided here.
Since 1986, the Riemann-Hilbert problem has been shown not to be solvable in general and a counterexample is provided.
Appendix six gives an account of the Picard-Vessiot theory (a Galois theory for ODEs) up to the early 1900s.
Appendix seven presents the work of Appel and Goursat on the hypergeometric equation.
The point of view of Poincaré for solving ODEs is developed and implications are highlighted. Three texts of Poincaré (1880) are commented.
This book can only enrich a scientific library.