Lessons I wish I had learned before I started teaching differential equations [pdf] (1997)

One of the most interesting applications of ODEs (in my opinion) is the field of control theory, which empowers autonomous systems/robots/vehicles to guide themselves. Brian Douglas has some fantastic introductory videos on the subject [1]. I don't think I agree completely with section 5 (Forget About Existence and Uniqueness of Solutions). For an ODE in the form xdot = f(t,x), existence and uniqueness on an interval follows if f is continuous in t and Lipschitz continuous in x [2]. This…

There seems to be something similar in other branches of mathematics too (and lots of other fields). A clear example I recall was studying Fourier transforms in math and I couldn't make any sense of it. To me it was just some by-rote algebra with integrals of exponentials of complex numbers. But then I happened to do some audio signal analysis, and when I saw a magnitude spectrum of a waveform, it was instantly obvious what's going on (and understanding…

I TA'd for an undergrad ODE course for two semesters. I'm a grad student at a public university with very good undergrad engineering students. Nonetheless, with the course I taught, at least, the main problem was that WAY too much was packed into a single semester. I suspect that this is the same everywhere. For example, about a month and a half of the course was devoted to systems of linear equations. These were students who, a priori, knew basically…

Some thirty or so years ago, Bessel functions were included in the syllabus, but in our day they are out of the question. lolwut? Maybe it's just a physics thing, but most of the differential equations we solved in college needed Bessel functions of some sort (usually because of cylindrical/spherical symmetry).