Runge-Kutta methods are designed to approximate Taylor series methods,
but have the advantage of not requiring explicit evaluations of the
derivatives of . The basic idea is to use a linear combination
of values of to approximate . This linear combination
is matched up as closely as possibly with a Taylor series for
to obtain methods of the highest possible order **p**.
Euler's method is an example using one function evaluation.

We illustrate the development of Runge-Kutta formulas by deriving a method using two evaluations of per step; the technique employed in the derivation extends easily to the development of all Runge-Kutta type formulas. Given values , , choose values , and constants , so as to match

with the Taylor expansion,

as closely as possible. In what follows all arguments of **f** and its
derivatives will be suppressed when they are evaluated at .
It will also be convenient to express ,
as