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The simplest method of solving ODE's is Euler's method:
In order to integrate from the initial data at t = 0 up to
t = T, divide the interval into N equal steps of size
h = T/N and approximate y(tn = nh) = yn.
This method works and is sometimes used in
practice, but much better alternatives are described below. Nonetheless,
it is the conceptual basis of all other methods, and a little
analysis gives insight into how they all work.
G. Bard Ermentrout