Difference equation my answer different from books

[fahraynk]

Homework Statement

When drugs are used to treat a medical condition, doctors often recommend starting with a higher dose on the first day than on subsequent days. In this problem, we consider a simple model to understand why. Assume that the human body is a tank of blood and that drugs instantly dissolve in the blood when ingested. Further assume that the drug vanishes from the body at a rate that is proportional to drug concentration.

Let X[n] represent the amount of drug taken on day n, and let y[n] represent the total amount of drug in the blood on day n, just after the dos x[n] has dissolved int eh blood, so that :
y[n]=x[n]+##\alpha##[n-1].

Assume that no drug is in the blood before day 0, and that one unit of drug is taken each day, starting with day 0. Determine an expression for the amount of drug in the blood immediately after the dose on day n has dissolved.

Homework Equations

The Attempt at a Solution

I plug ##y=P^n## into the difference equation to get ##y_h=K\alpha^n## and I plug ##y=C## into the equation for the particular solution and get for an answer :
## y = K\alpha^n - \frac{x(n)}{\alpha-1}##

The book says this:
Solve by iteration :
##n : y[n]##
##0 : 1##
##1:1+\alpha##
##2:1+\alpha+\alpha^2##
##3:1+\alpha+\alpha^2+\alpha^3##
##...## : ## ...##
##n : \frac{1-\alpha^{n+1}}{1-\alpha}##
Books answer is : ##\frac{1-\alpha^{n+1}}{1-\alpha}##

I think the book is using a power series representation or something like that. Is my answer wrong for x(n)=1 ? If so what am I doing wrong ? I rather not use a power series if I can answer the difference equation using methods I already know like undetermined coefficients.

 

[S.G. Janssens]

y[n]=x[n]+α[n-1].

Do you mean: ##y[n] = x[n] + \alpha y[n-1]## where ##\alpha \in (0,1)## determines the rate at which the drug disappears from the blood?

I think the book is using a power series representation or something like that.

It is calculating the first few terms of the solution, recognizing a geometric series and then using the standard formula for the partial sums of such a series.

 

[vela]

I plug ##y=P^n## into the difference equation to get ##y_h=K\alpha^n## and I plug ##y=C## into the equation for the particular solution and get for an answer :
## y = K\alpha^n - \frac{x(n)}{\alpha-1}##

I'm not sure why you're writing the solution in terms of x(n) since you've assumed x(n) is a constant to find the particular solution. Just write the constant.

You didn't finish solving the problem. You must still determine the arbitrary constant K.