Differential equations

Now that I've graduated, I'm back to teaching. This summer, I'm teaching the venerable Applied Differential Equations I. The fun began when I started trying to create a schedule, and realized that the list of topics to cover refers to section that no longer exist in the new version of the text. For example: according to the Master Plan, I am to teach sections 1.4 (the phase plane) and 1.5 (Euler approximation). In reality, the former edition's 1.5 has been moved to the new edition's 1.4, and the former edition's 1.4... doesn't exist at all as a section! Instead it's a group project at the end of chapter 1, with no homework problems at all.

Sigh. In any case, I'm in agreement with my advisor that the text lays out the subject without a logical flow; rather, it strings a mishmash of unrelated methods one after the other. So, I'm reorganizing and adapting different topics here and there, with the hope that the students will better see how the topics fit together logically and flow sensibly, starting with their understanding what the name of the class actually means.

For example: the text (and most ODE texts, I believe) teaches the solution of first-order linear ODEs in section 2.3, and advises the use of integrating factors. Second-order linear ODEs are treated in chapter 4, with several methods presented, none of which involves an integrating factor. Of course, only one of them works all the time: solving a homogeneous system, then (if necessary) applying variation of parameters. Of course, this method also works for first-order linear ODEs. Why do we teach two different methods? beats me.

So, I have banished integrating factors as a method of solving first-order linear differential equations; moreover, I will present the method of undetermined coefficients until after students have practiced solving second-order linear ODEs using the exact same method that they used for first-order linear ODEs.

If that sounds scary to you, fret not: it scares the bejeezus out of me, too. And I'm teaching the thing.

The trouble, alas, is that I don't ordinarly do differential equations. My research is algebraic, and it's so algebraic that I haven't touched a differential equation since I taught the same course three summers ago. The result is that I'm spending an awful lot of time in the office preparing notes, checking to make sure that I can do the homework (thus answer homework questions, should they arise), and so forth.

Meanwhile, I'm also choosing a book for the fall class that I'm teaching at North Carolina Wesleyan College (where I have been hired — thanks Alessandra for mentioning Chronicle Careers): Linear Algebra. I will restrain myself to saying that the vast majority of linear algebra books are about as bad as the differential equations texts: a collection of the author's favorite elementary methods and applications, rather than a simple, coherent, focused idea explicated as simply as possible.

Alas, the few books that do seem to meet my standard, are above the level of the students I expect to have, and (on top of it) tend to start with vectors instead of where they should start: linear equations. There's a reason it's called linear algebra, and in my not-so-humble opinion, all math texts should begin by explaining their subject matter.

That's how I started my dissertation: by explaining the title (Combinatorial Criteria for Gröbner Bases). Admittedly, it took 60 pages, but I started with a high-school problem and worked my way up. :-) The text itself is 260 pages, so taking 60 pages doesn't seem so bad to me.