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# Differential equations: course

### From Mathematics Is A Science

Planned for Fall 2011.

## 1 Description

This is a one-semester course in differential equations and applications.

## 2 Prerequisites

## 3 Outline

As far as modeling, here's what I have in mind. Typically, this is how things go in such a course:

model -> derivation of the DE -> analysis -> discretization -> difference equation -> -> simulation -> analysis etc.

I would like to start with discrete differential forms (you can see some of it here: Discrete exterior calculus). Then instead of DEs, one will deal with "discrete differential form equation" (I don't know a better name yet). Those are something like a more sophisticated version of cellular automata. Then this how it will go:

model -> derivation of the discrete differential form equation -> simulation -> -> analysis -> continuous differential form equation (same!) -> DE -> analysis etc.

The modeling will be done with Excel.

Topics:

- Review of calculus
- First-order ordinary differential equations
- Second-order ordinary differential equations
- Differential forms
- Applications: vibrations and electrical circuits
- Laplace transform, approximate solutions, orthogonal functions
- Fourier series
- Partial differential equations: heat, wave, and Laplace equations

- Differential equations: exercises
- Differential equations: test 1
- Differential equations: test 2
- Differential equations: midterm
- Differential equations: final exam