Admittedly a soft question but an important one, I think. The questions I've asked below are questions that can be answered, and not just discussed.
I read this essay yesterday by Gian-Carlo Rota denouncing the structure of most differential equations courses. In the essay, he says that the structure of these courses hasn't changed since the 19th century, almost word for word in some cases, and is filled with redundant lessons.
In particular, he has a problem with the teaching of exact equations, integrating factors, homogeneous differential equations, and existence and uniqueness of solutions, saying that all of these topics and techniques are of no use whatsoever.
He believes the bulk of DE courses should be linear DEs with constant coefficients (not variable coefficients), linear algebra, and Laplace transforms.
Are most people here in agreement with this? If so, how would a better DE course be built from scratch?
EDIT: I just looked at MIT's syllabus for Differential Equations, and it looks like it reflects what Rota said. So there's truth in it, though I haven't checked yet whether or not the course text by Edwards & Penney follows suit.
Best Answer
I can sort of see what Rota is arguing for but it's kind of a fine line because his arguments should also apply to calculus: "why teach epsilon-delta methods", "why teach volume integration of ridiculous rotated surfaces" etc etc. More to the point, "why teach abstract nonsense to applied-math majors?" I would argue that, aside from giving students an introduction to differential equations, such a class should primarily serve as a reinforcement of recently learned mathematics (see below).
I think Rota's main point is that we shouldn't teach non-linear differential equations in introductory classes. For me, the biggest positive about differential equations as they are taught, is that they give students one of the first direct applications of linear algebra. You really get to see the power, and the abstract usefulness of linear algebra and how it applies to things that students previously didn't think of as vector spaces. It reinforces the power of thinking about kernels, eigenvectors, jordan-block decompositions, etc. I eagerly support differential equations being taught in tandem with linear algebra.
As well, numerical differential equations go a long way to reinforce ideas from calculus like infinitesimals, taylor approximations, etc. So I would emphasize numerical methods more. This is primarily because we cannot solve exactly 99.9999% of differential equations out there explicitly. Moreover, numerical methods would show the importance of understanding basic linear differential equations and how they describe, to first order, the nature of non-linear equations.
On the subject of linearity, teaching first order coupled equations isn't bad either, showing the basic fixed point theory and a very brief overview of nonlinear behavior .
If you're a physicist, you'll learn Sturm-Liouville theory, Bessel functions and all that jazz in a later "math for physics" course. If you're a mathematician, you'll learn the finer details of uniqueness, wronskians and nonlinear dynamics later on.
Here's a caveat though. On the subject of reinforcing knowledge, teaching integrating factor methods and similar techniques goes a long way to improve student's understanding of calculus, specifically integration and differentiation. So I wouldn't completely get rid of that either.