 
Summary: NUMERICAL METHODS FOR ODEs
Consider the initial value problem
y0 = f(x; y); x0 x b; y(x0) = Y0
and denote its solution by Y (x). Most numerical
methods solve this by nding values at a set of node
points:
x0 < x1 < < xN b
The approximating values are denoted in this book in
various ways. Most simply, we have
y1 Y (x1); ; yN Y (xN)
We also use
y(xi) yi; i = 0; 1; :::; N
To begin with, and for much of our work, we use a
xed stepsize h, and we generate the node points by
xi = x0 + i h; i = 0; 1; :::; N
Then we also write
yh(xi) yi Y (xi); i = 0; 1; :::; N
EULER'S METHOD
Euler's method is de ned by
yn+1 = yn + h f(xn; yn); n = 0; 1; :::; N 1
