Example 9.11.  Given the repeated dosage drug level model   with the initial condition  .
9.11 (d).  Use convolution to find the solution.

Explore Solution 9.11 (d).

The solution to the homogeneous equation     is   .

The transfer function is    and the unit-sample response is   .

The input sequence is    and it's z-transform is   .

The particular solution is calculated with the formula    as follows:

Details

In convolution form    and we have

        

Remark.  The particular solution    obtained by using convolution has the initial condition  .  

The total solution to the nonhomogeneous difference equation is  

          

          

Now we compute  ,  and solve for the constant    which will produce the proper initial condition.  Therefore,

         ,  

which can be arranged to yield   .  

We can explore the situation when  ,    and  .

Figure 9.1.  The solution to with .

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