## Reports for differential equations course

This coming academic year I am scheduled to teach differential equations. I’ve taught this course each year for the past several years, and each time have had a written paper due at the end of the term. In that paper students have to analyze a certain non-linear system using all the various methods developed throughout the course.

One difficulty I’ve had with this assignment is giving feedback to the students. In the past I’ve had drafts of the paper due at various points, and given them some comments on their drafts. This has not been an effective way to develop their technical writing; nor has it been effective at getting them to use LaTeX well.

This year I am going to try assigning a series of shorter papers, rather than one long paper. The idea is that each will focus on only one or two differential equations topics; each will also focus on only one or two “technical issues.” These “technical issues” involve using LaTeX for typesetting, and Mathematica for generating plots, running basic simulations, etc.

(Eventually, I’d like to move away from Mathematica towards something like Python, but that’s another story.)

Here is a possible list of paper topics and goals:

1. In the first paper the students will qualitatively analyze an ODE of the form $u^\prime = f(u)$. They will need to get Mathematica to plot the function $f$ as well as the slope field. They will analyze the ODE by finding the equilibria, determining their stability, etc. Part of the mathematical content they will be learning is to analyze stability from the plot of $f$ (which I won’t talk about in class), rather than from the slopefield (which I’ll do in class). There will be a large emphasis in this first paper on using LaTeX well.
2. For the second paper, the focus will be Euler’s method. The goal will be to write an expository paper about the method, complete with illustrations.
3. The third paper will come when we are studying linear systems and using those tools to analyze equilibrium points for non-linear systems. I’ll need to come up with a good system for them to analyze, and then have them write a report about this system. In principle, it would be nice to have some of the analysis be part of their regular homework. In class, many of the examples are predatory-prey models. Perhaps this paper can discuss something different, such as a SIR disease model. (Then I can assign a nullcline analysis as a bonus problem.)
4. The final paper will concern conserved quantities. Probably I will give them a potential energy curve for some system and ask them to analyze the possible solutions, characterized by their energy level. Orbits (of a certain fixed angular momentum) around a Schwartzchild black hole are one possible system.

In order for this series of assignments to achieve the goal of improving LaTeX, Mathematica, and writing skills, I will need to develop a good grading rubric. I will also need to come up with a good list of writing tips and guidelines.