I have taken a lot of math in university, but chose to omit differential equations. Unfortunately, now I have to read computer science proofs that use them, mostly ODEs, and this is always a struggle. What textbook(s) should I read to take me from the basics to practical use of the theory?
If you are looking for practical solving of differential equations I recommend :
Advanced Mathematical Methods for Scientists and Engineers by Carl M. Bender and Steven A.Orszag.
Methods of Mathematical Physics R.Courant and D. Hilbert two volumes.
The second positions is absolutely the best ever written textbook on differential equations.
Maybe not so practical for solving special types of ODE's but still close to applications is
- Ordinary Differential Equations by V.I.Arnold
Another quite well written textbook dealing with differential equations is
- Dynamical Systems with Applications using Mathematica by Stephen Lynch
This book however mainly describes qualitavie methods for dynamical systems, but there are many good examples.
Boyce and DiPrima, Elementary Differential Equations, Eighth Edition. Website.
The books which I referred are:
Ordinary differential equations by E. Coddington.
Differential Equations With Applications and Historical Notes by George F. Simmons. This book is a classic and at the end of each chapter has historical notes of Mathematicians which is fun to read.
An online book published by the AMS is available here. I basically learn't the proofs of Gronwall's equality from here and used it as a reference book.