I am tasked with teaching an introductory differential equations course. In my experience, this course tends to be a laundry list of methods that apply to specific forms of differential equations where each form needs to be memorized. I think such a course lacks continuity and appeal, so my question is: what are possible themes for a first differential equations course?

Some ideas I have come up with are:

  • Modeling. The course should be focused on taking real-life situations and producing differential equation-based models. I suspect most of these models would be unsolvable, so maybe this course would also have to focus on numerical methods.
  • Reversing Calculus. Cast each method as undoing a method from regular calculus. This would work for several of the methods, but maybe not for things like Laplace transforms...
  • We can solve very little. The focus of a course could be on the fact that most equations we cannot solve, but there a couple of types that we can, so we should work to approximate the equations we cannot solve with ones that we can.
  • Change of basis. Make the course about the Laplace transform and the Fourier transform and converting back and forth between calculus problems and algebra problems. I don't know whether this subsumes all of the traditional techniques or not...

I have very limited experience with differential equations, so I'd appreciate comments from people with a broader view of the subject—I could be missing the point entirely!

Note: I am teaching the only section of first-year honors diffeq, so I have some flexibility. The students are studying sciences, so mostly physics, chemistry, and biology. A few of them are math majors.

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    If you have little experience with differential equations, I suggest finding a book (perhaps from a list provided by your curriculum) and following it. That gives a safer ground to build on and things can't fail that badly. Your school should have some guidelines as to what the students are supposed to learn, and you shouldn't deviate too much. – Joonas Ilmavirta Dec 2 '15 at 16:30
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    Whatever you do, don't forget that the graduates of your course will have to be able to do that laundry list for their future courses. – Gerald Edgar Dec 2 '15 at 19:03
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  • Personally, I like the standard calculations. Rather than avoid them I embrace them, but, to make it interesting, try to see deeper math in the methods. Don't just teach exact equations, think about Pfaff's theorem. Or, rather than just solve a system of ODEs, find the integral curve of a vector field. In summary, one could teach a course centered around differential forms and vector fields with the intention of introducing students to these objects in the nice context of flat space. If done right you could both cover the standard technique and give them a deeper geometric direction 4 future.. – James S. Cook Dec 3 '15 at 18:28
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    "We can solve very little." I think this is a great theme! Gives students that rare panoramic view. – Joseph O'Rourke Jan 3 '16 at 1:00

These are some of the questions I would use to guide the course:

  • What is a differential equation ? In particular, what language should we use to communicate the structure of a differential equation.
  • What is a solution to a differential equation ? In particular, what object is best suited to capture our intuitive concept of a solution?
  • Can we free ourselves of coordinates in both of the items above?
  • For each example we study, what is the relation between different solutions? What is the structure of the general solution?

I have not fully embraced this program in my teaching, it would be difficult with the general audience of the course. But, I think there is room for creative implementation of symmetry methods in the elementary DEs course. I want someone to write the book which does this, while not sacrificing standard techniques.Of course in addition, it seems obligatory to add:

  • where do differential equations appear in other math and the real world

This question is actually the easiest I've listed.

I'll just give a rough description of my first ODE's course, take from it what you will!

Luckily we had lessons split up into theory and problem solving, so during the entire first month the problem solving classes were solely dedicated to liquidating the laundry list you mention (basically 4-5 known quadratures for 1D equations). During this time, the theory lessons were doing something completely different.

At first the theory was just an introduction to the "ODE problem": order, dimension, normal form, IVP, etc. Then we studied fundamental theorems. These were Peano's theorem, the Picard-Lindelöf theorem, and regularity of flow. We finished this at the same time that the problem lessons finished with the laundry list.

From then on problem lessons followed what theory was doing, just with specific instances of ODE's. So I'll just describe the theory now. We took up the topic of first order linear IVP's (in any dimension): $\exists !$ for all time, fundamental matrices, vector space structure of solutions, Gronwall's inequality (in general, then applied to linear), Liouville's theorem, the exponential of a matrix (and how to take advantage of Jordan classification), and finally classification of critical points (I say it as one phrase but of course it took weeks). The next topic we did was sort of an indulgence of our teacher who specialized in it. It was on non-linear periodic ODE's, so things like the monodromy matrix, stroboscopic map, etc. were looked at. Lastly we looked at (what little can be said of) nonlinear ODE's. I think a good summary would be that the culmination of this topic was for us the Hartman-Grobmann theorem.

That's about it, I think. As for what to take from this, I'd say the most important tool for a general audience is a good understanding of linear ODE's, even moreso than the laundry list for 1D order $n$ ODE's. I've continually appreciated in later courses the knowledge (practical and theoretical) of linear ODE's. And it's not hard to motivate: the only damn thing we know how to do is linearize problems, right?

  • what a beautiful course. What were the prerequisites? – James S. Cook Jan 28 '16 at 23:25
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    I enjoyed it! Prereqs were linear and multilinear alg., differential and integral calculus (in $\Bbb R ^n$), topology, physics (for examples), computer science (not sure why - we didn't do anything numerical), and some complex analysis. From: fme-intranet.upc.edu/tmp/consgd/2015/200141-e-3.pdf – GPerez Jan 29 '16 at 2:51
  1. I don't think you understand what the students need. You seem like a math type, not a scientist, not a math teacher serving scientists. Not meant adversarially, honest. Consider this a blind spot.

  2. I would be hesitant about iconoclastic courses (not a laundry list, different from traditional) given (1). You may make it worse.

  3. I would go with reversing calculus among your choices. Just looking at a diffyQ, you see a bunch of derivatives. It is similar to integral calc, which students have had. Note, that this basically means a set of techniques, similarly to integral methods in calc 2.

  4. I would avoid modeling since kids who need it will get it in required classes. Word problems are hard. Plus ther is limited time in ODEs to hit various forms, just as math. Let them learn the basic second order constant coefficient nonhomo as an algebraic form. Then when they encounter it in a myriad of places (e.g. controls), they will at least feel they know the math, while dealing with the physical concepts (how are inductors and capacitors different; dashpots versus springs; etc). Also, I don't think you have time to do justice to word problems as well as normal ones.

  5. No on we can't solve a lot. They will learn that soon enough. And they need analytical solutions for lots of derivations (e.g harmonic oscillator). There is a difference between designing a specific engineering model and derivation of broad equations in fluids, thermo, physics, chemistry, etc.

  6. No on change of basis. Too hard, too end of course to be a theme for an intro class.

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