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.