Exercise 19.  Consider the system of difference equations

with the initial conditions   .  

19 (b).  Use z-transforms and construct the solution in part (a).

Solution 19 (b).

Method (i)   Take the z-transform of both sides and use the initial conditions   :  

                          
                    
then get  

                          

which can be written as

                      

Solve for and and get

                       
                    
Then get

                       

Use Table 9.1 to find the inverse z-transform   

                    ,  
                    and  
                    .  

We are done.   

Method (ii)   Using residues we obtain

                      
                    
                    and
                    
                      

Using residues we can find  .  

                      

The other solution is found in a similar fashion.  

                      
and
                      

Using residues we can find  .  

                      

Therefore,

                    ,  

                    and   

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

Using limits and residues we obtain    

Using limits and residues we obtain    

          The Maple commands are similar  

  

                                                            

Using limits and residues we obtain    

  

                                                            


  

                                                            


  

                                                            

Using limits and residues we obtain    

  

                                                            


  

                                                            


  

                                                            

We are really done.   

Aside.  We can use Mathematica's InverseZTransform subroutine.

          The Maple commands are similar  

  

                                                            


  

                                                            

We are really really done.   

Aside.  We can use Mathematica's Rsolve subroutine.

          The Maple command is similar  

  

                                                            

        It takes some effort to see that Mathematica's solution is equivalent to the analytic answer.

                    

                    

When    is even we have

                      

                      

When    is odd we have

                      

                      

We are really really really done.   

Aside.  We can graph the solution.

                                                                                                                        The sequence   .   

                                                                                                                        The sequence   .   

                                                                                                                        The sequence   .   

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell