Exercise 4.  Solve the homogeneous difference equations.  

Exercise 4 (a).     with   .   
                          Hint.  Get  .  

Solution 4 (a).

See text and/or instructor's solution manual.

Answer.   .  

Remark.   It is our goal to learn Z-transforms and use residues.  

Solution.   

Method 1.  The characteristic equation  

                      

has roots and .

The general solution is  

                    .

Solve the linear system  

                      

and get    and  .

Therefore,  

                    .

Method 2.  Take the z-transform of both sides and use the initial conditions   :  

                    ,  

and then get  

                    .  

New solve for    and obtain:  

                    .  

Using Tables.   Using Table 9.1 and the formula      we get:  

                      

Remark.  The details for the partial fraction expansion are at the bottom of the page.

Therefore,  

                    .

We are done.   

Alternative Solution Using Tables.   Using ordinary partial fractions and Table 9.1  

and the formula      we get:  

                      

        Since    and   we can write  

                       and   

                       for   .  

Therefore, the solution has the following form:

                       

We are done.   

Aside.  The commands for the ordinary partial fraction expansion are:  

        Here      and   ,   and we can corroborate this solution.

We are really done.   

Using Residues.   Calculate the residues    at the poles   .  

                      

                      

Therefore,  

                      

We are really really done.   

Aside.  We can let Mathematica double check our work.

Aside.  The Maple commands are similar  

  

                                                            

  

                                                            
                                                            
                                                            
  

                                                            


  

                                                            

We are really really really done.   

Aside.  We can use Mathematica's Rsolve subroutine.

Aside.  The Maple command is similar  

  

                                                            

We are really really really really done.   

Aside.  We can graph some of the terms in the sequence.

                    The sequence   .  

We are really really really really really done.   

The Details for the Partial Fractions.   

Aside.  How can we expand      into the proper partial fractions?

It is natural to use the standard partial fraction expansion and the command:

However, as we have seen, this will produce a solution involving the    functions.   

This can be overcome if we use a special partial fraction expansion that is easier to use with Table 9.1.

Method (i).   Use the following algebra steps  

                      

Method (ii).   Find the linear combination of   ,  

                    .  

Equate the numerators   ,  

and solve the linear system  

                    

and get   .   

Therefore, the desired partial fraction form is  

                    .  

Aside.   The Mathematica commands for Method (ii)  are

Method (iii).  (For distinct real roots)   First make the substitution      in      and get  

                    .  

Then use the standard procedure for expanding in partial fractions   

                    .  

Then make the substitution      in      and get  

                      


Therefore, the desired partial fraction form is  

                    .  

Aside.   The Mathematica commands for Method (iii)  are

Now we have the desired partial fraction form:

                    .  

This solution is complements of the authors.

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