1 Chapter 5 DIFFERENCE EQUATIONS

2 WHAT IS A DIFFERENCE EQUATION? A Difference Equation is a relation between the values y k of a function defined on a discrete set of arguments x k. In general a difference equation involves differences of a function. We always assume that the arguments are equally spaced.

3 PRELIMINARIES The Order of a difference equation is the difference between the largest and the smallest arguments k appearing in it. For example, the equation y k +1 = a k y k + b k is of order 1 and y k+2 – 5y k+1 + 6y k = 0 is of order 2. A Solution of a difference equation will be a sequence of y k values for which the equation is true, for some set of consecutive integers k.

4 WHAT IS A LINEAR DIFFERENCE EQUATION? A general Linear Difference Equation of nth order is of the from (a 0 E n + a 1 E n-1 + a 2 E n-2 + … + a n-1 E + a n ) y k = f(k) where a 0, a 1, a 2, …, a n and f(k) are given functions of integer argument k. If f(k) = 0 then the equation is called a Homogeneous Linear Difference Equation of order n. Otherwise it is called Non- homogeneous linear difference equation.

5 SOLUTION FOR HOMOGENEOUS LINEAR DIFFERENCE EQUATIONS WITH CONSTANT COEFFICIENTS

6 Case 1

7 Case 2

8 Case 3

9 NON-HOMOGENEOUS LINEAR DIFFERENCE EQUATIONS WITH CONSTANT COEFFICIENTS

10 Particular Solutions Depends on f(k) Simple cases

11 Solve y n+3 – 6y n y n+1 – 6y n = 3 n Homogeneous solution y n = C 1 + C 2 2 n + C 3 3 n Particular solution

12 Solve (E 2 – 5E + 6) y n = 4 n (n 2 – n + 5)