Calculus: Concepts and Methods

Difference equations and the shift operator E are introduced in this chapter. The chapter is mainly concerned with higher order linear differential and difference equations with constant coefficients. The analogies between the two types of equation are exploited by showing that essentially the same techniques work for both differential and difference equations. For this reason, the symbol x is used to denote the variable in both types of equation, though in the case of difference equations, it only takes discrete values, which are nonnegative integers. The techniques are easy, provided some mastery of complex numbers has been garnered from the previous chapter. A brief theoretical justification for the techniques described appears in §14.3 and §14.4, which will be of particular interest to those with a knowledge of linear algebra. But readers impatient with theory will find it adequate to confine their attention in these sections to the manipulation of operators as described in the examples of §14.1 and §14.2 and to the assertions of §14.3 and §14.4 without delving into the reasons why these are correct.

It should be noted that the technique for finding particular solutions of nonhomogeneous equations described in §14.7 is only one of several. We have chosen to present this technique because it involves no essentially new ideas. But it is often quicker to find particular solutions using operators in a more adventurous way than described here.

In §14.8 there is a short discussion on the stability of the solutions of equations.