Expanding the solutions of implicit sets of ordinary differential equations in power series

Abstract

Given a general analytic set of implicit ordinary differential equations the method described here will find an algorithm for expanding solutions to the equations in power series. The method is entirely mechanical, and does not have to rely on its user for prompting about the order, degree or form of the equations. In simple cases, such as sets of equations that can trivially be reduced to explicit form, the algorithm generated will be that of the Taylor Series method of Barton, Willers and Zahar (1972a,b) and so it can generally be expected to form the bases for effective and efficient numerical procedures.