In the OP, the author Gian-Carlo Rota
started out with:
> 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.
> 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.