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2.2.2 Runge-Kutta Methods     continued...

So the object is to match

with the Taylor expansion

Equating coefficients of like powers of h in the above expressions for R and T, we are able to obtain agreement in terms involving and :

If we choose , an arbitrary parameter, these equations can be solved exactly to give

Combining all of this gives a one-step method of order p = 2 if and f is sufficiently smooth. We state this in the following algorithm.