In preparing this course, there was a serious question of which textbook to
use. Ordinary Differential Equations is THE oldest area of modern applied
mathematics, I would argue, and as such, has a very deep and broad accumulation
of results. In particular, the style of material taught in the course has
changed considerably over the last 100 years, as new technologies and results
have emerged. Unfortunately, this means that many important and valuable
results from the past can no longer be fit into a single-semester course on the
subject.
Dedicated students, then, should be made aware that there are more results
that they may someday find use for. Many of these results are found in
older and alternative textbooks. Here's a list.
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Building on
Elementary Differential Equations by William E. Boyce and Richard C. DiPrima
Main textbook, which is oriented towards studying nonlinear dynamic systems
Differential Dynamical Systems by James D. Meiss
Secondary textbook
An Introduction to Ordinary Differential Equations, by R. Agarwal and D. O'Regan
Classic textbooks for Upper-level Ordinary Differential equations
Ordinary Differential Equations by Garrett Birkhoff and Gian-Carlo Rota
Ordinary Differential Equations by George F. Carrier and Carl E. Pearson
Theory of Ordinary Differential Equations by Earl Coddington and Norman Levinson
Ordinary Differential Equations by E. L. Ince
Supplemented by
Advanced Mathematical Methods for Scientists and Engineers by Carl M. Bender and Steven A. Orszag
for an introduction to approximation methods