# Category Archives: Differential equations

## Using Overleaf for student papers

In my introductory differential equations course, I assign students to write several papers. I require the students to typeset their papers with LaTeX, and to use graphics imported from Mathematica. This semester, the paper assignments are: Equilibrium solutions for the … Continue reading

## Mathematica — Euler’s method for systems, revisited

An earlier post discusses Mathematica code for Euler’s method. Here is some updated code. A couple preliminary notes: I like to clear all variables at the beginning. For systems it is fun to superimpose the plot on top of the … Continue reading

## Oberwolfach report: Weakly asymptotically hyperbolic metrics and the constraint equations

Earlier this month I gave a lecture at Mathematisches Forschungsinstitut Oberwolfach as part of participating in the workshop “Mathematical Aspects of General Relativity.” My Oberwolfach Report contains a short description of the contents of my lecture.

## Math 235, Spring 2015: Comments on Report 2

After grading (most of) the second round of reports, I have the following comments: While it is necessary to state the methods being used to analyze a system (such as linearizing about an equilibrium), it is not necessary to explain … Continue reading

## Math 235, Spring 2015: Comments on Exam 1, Problem 1

In this problem, students were first asked to verify that was a solution to the differential equation and to plot that function. Remember when verifying that a function is a solution to a differential equation that one cannot assume that … Continue reading

## Comparing numerical methods

Here is a simple example of comparing different numerical methods for ODEs. We focus on the example subject to the initial condition . We know that the exact solution is . Let’s compare Euler’s method, the trapezoid method, and the … Continue reading

## Mathematica – Euler’s method for systems

Here is a description of how to implement Euler’s method for a system. We focus on a simple example: where and First, let’s put in the functions which define the system Next, we make a StreamPlot of the vectorfield Now … Continue reading