Eulers method in general is just a taylor expansion of a function involving the initial condition and the derivative.
We know that the first two taylor series terms are f(x) = f(a) + (x-a)*f'(a).
If we have a system of equations, we replace x's and a's with vectors where f'(a) is a linear object (i.e a nxn matrix for system of n equations) and our Euler up-date becomes:
f(x) = f(a) + C*(x-a) where a = [a1,a2,...,an] and x = [x1,x2,....,xn] and C is nxn derivative evaluated at vector a.
What computational tools do you have? Do you have MATLAB or the open source version Octave (and GUIOctave)?