What a programmer needs to know about ordinary differential equations

[What are ordinary differential equations? Read my detailed notes here on Evernote:Math:ODE]

Solving an ordinary differential equation is computationally expensive. One of the most common approximation methods used to solve this is euler. Understanding this for non math buffs means a lot of effort.(Lets say "Ordinary.Differential.Equations_Tenenbaum_Pollard" has too many pages). So I wrote a python program that implements both euler and its more accurate form the predictor corrector method and thought I will share it.

What are we trying to achieve?

1. Implement euler and corrector predictor method
2. Plot them on a graph to understand their nature
3. Compare the graph plots to visually understand the accuracy differences

What equation are we trying to solve?

dy/dx = x + 2y

What python libraries do we need?

#http://matplotlib.sourceforge.net/

import math

from pylab import *

Show me the code?

http://dl.dropbox.com/u/46029347/ode.py

What are the results?

We can visually see from the following plots that

1. Reducing step size increases accuracy
2. The predictor corrector (shown with '+' ) is more closer to original equation (shown with '.' ) than the traditional euler method (shown with '*')

When we plot from 0 to 3 with step size 0.2

When we plot from 0 to 3 with step size 0.1

When we plot from 0 to 3 with step size 0.01

-- Python
-- Pure awesomeness :)