Example 9.10.  Solve the difference equation    with initial condition  .
9.10 (a).  Use the z-transform and Tables 9.1 - 9.2 to find the solution.
9.10 (b).  Use residues to find the solution.

Solution 9.10.

Solution 9.10 (a).  Take the z-transform of both sides  

            .

Solve for    and get  .  Then expand and obtain

            .  

Find the inverse z-transform of each term

            

When    we get  ,  and when the expression for simplifies to be

            

Solution 9.10 (b).  Start with the formula   in part (a).  Then use the Corollaries 9.1 and 9.2 and residues to find  .  

                

(c) 2006 John H. Mathews, Russell W. Howell