Page 342 - Schaum's Outline of Differential Equations

CHAPTER 34

An Introduction to

Difference Equations

INTRODUCTION

In this chapter we consider functions, y n =/(«), that are defined for non-negative integer values n = 0, 1,2,
3
y
3, ... So, for example, if y n = « -4, then the first few terms are {y 0, ^, y 2, y$, y^, ...} or (-4,-3,4, 23, 60, ...}.
Because we will be dealing with difference equations, we will be concerned with differences rather than
derivatives. We will see, however, that a strong connection between difference equations and differential
equations exists.
A difference is defined as follows: Ay n = y n+\—y, and an equation involving a difference is called a difference
equation, which is simply an equation involving an unknown function, y n, evaluated at two or more different n
2
values. Thus, Ay n = 9 + n , is an example of a difference equation, which can be rewritten as y n+\—y n = 9 + n 2
or

We say that n is the independent variable or the argument, while y is the dependent variable.

CLASSIFICATIONS

Equation (34.1) can be classified as a first-order, linear, non-homogeneous difference equation. These
terms mirror their differential equations counterparts. We give the following definitions:

• The order of a difference equation is defined as the difference between the highest argument and the
lowest argument.
• A difference equation is linear if all appearances of y are linear, no matter what the arguments may be;
otherwise, it is classified as non-linear.

• A difference equation is homogeneous if each term contains the dependent variable; otherwise it is
non-homogeneous.

We note that difference equations are also referred to as recurrence relations or recursion formulas
(see Problem 34.7).

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