Differential equations are recurrence relations in APL

An mth-order differential equation y(m)= f(x,y,y’,...,y(])))) is itself a numerical recurrence relation that can be iterated to produce the Taylor series solution. The purpose of the APL in this paper is to demonstrate this principle in it’s simplest form, This is accomplished as a direct application of the automatic differentiation workspace developed in [10]. After a brief review of this workspace, we implement the principle for fwst-order, mth-order, and systems of differential equations. The shortcomings of this implementation lead to a discussion of more advanced techniques found in automatic differentiation literature. The general method can be used in efficient and accurate numerical software for solving differential equations. We begin with an explanation of how a differential equation can be considered a recurrence relation.