Ryan Smith

So let us take a look at the case where we need to obtain a numerical approximation to a ODE. Just go to wikipedia to find some theory of when and why this works…..i will show you the algorithm implemented in Mathematica.  The general solution looks like this:

consider the differential equation:

Mathematica Code:

Clear[x, y, x0, y0, h, n, k, k1, k2, tbl]
 
f[x_, y_] = y^2;
 
{x, y, h} = {0, 1, .2};
 
tbl = {{x, y}};
 
n = 6;
Do[
k1 = f[x, y];
k2 = f[x + h, y + h*k1];
k = (k1 + k2)/2;
y = y + h*k;
x = x + h;
AppendTo[tbl, {x, y}]
, {i, 1, n}]

So, your Mathematica notebook should look something like this.

While you are at it, you might as well plot the data: