Pursuing Analogies between Differential Equations and Difference Equations

The study of ordinary differential equations has long been a staple in mathematics at both the undergraduate and graduate levels. Recently, instruction in the study of difference equations has widened, primarily due to the expanded role of the digital computer in mathematics. The two topics are inextricably linked at all levels, from elementary techniques through current research questions. Pursuing the analogies between these fields of study can only deepen the understanding of each. In particular, the study of many elementary topics in difference equations, requiring not even the use of calculus, can serve as a founda- tion for intuition and understanding of the analogous topics in differential equations. Since typical difficulties encountered at the introductory level in studying differential equations include development of intuition and avoiding the approach of pure memorization of formulas, such a foundation is indeed useful. The purpose of this article is to illustrate how the analogy might be pursued through some typical problems and to comment on the usefulness of this analogy in the study of other topics. The comments here are, of course, not intended to be exhaustive; the goal is to suggest another way of thinking along with the traditional techniques for studying differential equations