Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]

Ten lessons I wish I had learned before teaching differential equations (1997) [pdf]
math.toronto.edu

> One of many mistakes of my youth was writing a textbook in ordinary differential equations. It set me back several years in my career in mathematics. However, it had a redeeming feature: it led me to realize that I had no idea what a differential equation is.

Wow! Good to see that he wrote this. Looking at his book,

Garrett Birkhoff and Gian-Carlo Rota, Ordinary Differential Equations, Ginn and Company, Boston, 1962.

I got the same impression! I couldn't see what the heck they were driving at. Instead, they seemed to flit around with a lot of tiny topics of little or no interest for little or no reason.

Want to understand ordinary differential equations, read Coddington:

Earl A. Coddington, An Introduction to Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1961.

Then for more, to make such equations much more important, read some deterministic optimal control theory, e.g., Athans and Falb

Michael Athans and Peter L. Falb, Optimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill Book Company, New York,

that, BTW, also has some good introductory, but very useful, material on ordinary differential equations.

More generally, want to know what to study in a subject that will be useful? Okay, one approach is to go to more advanced material that is an application of that subject and see what that material emphasizes for prerequisites, e.g., sometimes quite clear in an appendix.

E.g., Athans and Falb say quite clearly what is important in ordinary differential equations for their work.

Luckily, in this world, NSF grant writers are not typically undergraduate students, and there's plenty of money flowing towards ODE research (here are some examples: http://www.nsf.gov/awardsearch/showAward?AWD_ID=1600381&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1408295&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1318480&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1505215&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1418042&Hist... http://www.nsf.gov/awardsearch/showAward?AWD_ID=1346876&Hist...) and of course there is much more cash going into PDEs.

I don't get this. Differential equations theory is about proving existence and uniqueness of solutions. If you have to use numerical techniques to actually compute the solution, then that's perfectly fine. After all, even if the solution is explicit, like sin(x), or especially a special function, then we still need to use numerical techniques to actually evaluate that explicit solution.