Category Archives: Differential equations

Sage code for trigonometric forcing

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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

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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

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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.

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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

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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

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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

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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

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Euler’s method with Mathematica

We use Euler’s method to find an approximate solution to with initial value First, we set (The semi colon just suppresses the output.) Next we define the function Now we set up the recursion relation. We want the initial values … Continue reading

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Fredholm alternative and finite differences

I recently gave a talk at Seattle University describing a way to understand the Fredholm alternative via finite-differences. Slides related to the talk are here; an expository paper is here.

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