DifferenceEquationstoSection1.4DifferentialEquations.pdf

Difference Equations Section 1.4 to Difference Equations Differential Equations At this point almost all of our sequences have had explicit formulas for their terms. That is, we have looked mainly at sequences for which we could write the nth term as an = f (n) for some known function f . For example, if n + 1 an = n2 + 3 , then it is an easy matter to compute explicitly, say, a10 = 11 or a100 = 101 . In such 103 10003 cases we are able to compute any given term in the sequence without reference to any other terms in the sequence. However, it is often the case in applications that we do not begin with an explicit formula for the terms of a sequence; rather, we may know only some relationship between the various terms. An equation which expresses a value of a sequence as a function of the other terms in the sequence is called a difference equation. In particular, an equation which expresses the value an of a sequence {an } as a function of the term an−1 is called a first-order difference equation . If we can find a function f such that an = f (n), n = 1, 2, 3, . . ., then we will have solved the difference equation. In this section we will consider a class of difference equations that are solvable in this sense; in the next section we will discuss an example where an explicit solution is not possible. Example Suppose a certain population of owls is growing at the rate of 2% per year. If we let x0 represent the size of the initial population of owls and xn the number of owls n years later, then