Exercise 13.  In the exponential growth model    use the parameter   and initial condition  .

13 (a).  Use the z-transform and tables to find the solution.

13 (b).  Use residues to find the solution.

Solution 13.

See text and/or instructor's solution manual.

Answer.   .  

Solution 13 (a).  

        Substitute      into   ,  

and get the difference equation  

                    .

Take the z-transform of both sides and use the initial condition  :  

                    ,  

                    .  
        
Solve for    and get  

                    ,  

then use Table 9.1 to find the inverse z-transform   

                      

We are done.   

Solution 13 (b).  

        Using residues we get

                      

Therefore,

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

Take the z-transform of both sides.

Find the inverse z-transform.

          The Maple commands are similar  

  

                                                            


  

                                                            


  

                                                            

Aside.  We can use Mathematica's InverseZTransform subroutine.

          The Maple command is similar  

  

                                                            

We are really done.   

Aside.  We can use Mathematica's Rsolve subroutine.

          The Maple command is similar  

  

                                                            

Remark.  The solution can be written as  

                    .  

We are really really done.   

Aside.  We can graph the solution.

                                                                                                                        The sequence   .   

This solution is complements of the authors.

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