Abstract
The formal Taylor expansions for the exact solution to an initial value
problem can be expressed in terms of "Elementary Differentials". A similar
expansion for the approximation computed by a Runge-Kutta method enables
conditions for order to be written down. However, the necessity of these
conditions depend on the independence of the elementary differentials and
proofs of this are complicated and contrived. A special differential equation
will be discussed which separates the elementary differentials into distinct
components and provides a simple and natural proof of the independence result.
Other applications will also be discussed concerning the formation of
equivalence classes of Runge-Kutta methods.
