A short summary on the method:

1. construct an infinite sequence of numbers [f(0), f'(0), f''(0), ...] to represent a function f (note the the boundary conditions are specified directly)
2. the sequence can constructd by self-reference (e.g. y = y'), compsition (+, -, negate, *, diff), analytic function composition (g f = g fa :> g' f * f', for log, exp, trig)
3. By Taylor expansion, f(a+x) = sum(f[k] * x^k / k!), this allows us to evaluate f at any x.