# CHAPTER 2

# MODELING WITH FIRST-ORDER DIFFERENCE EQUATIONS

# 2.1. MODELING WITH FIRST-ORDER LINEAR HOMOGENOUS DIFFERENCE EQUATIONS WITH CONSTANT COEFFICIENTS

In this section, we investigate situations that are modeled by first-order linear homogenous difference equations with constant coefficients, that is, equations in the form

(2.1)

where *a* is a constant coefficient. We will iterate equation 2.1 with an initial condition to find a numerical solution, and we will also derive an analytical solution of equation 2.1.

**2.1.1. Model 2.1: Drugs**

Assume that the kidneys remove 20% of a drug from the blood every 4 h. Assume that the initial dose of the drug is 200 mg. Let *d*_{n} denote the amount of drug in the blood after *n* 4-h periods, and *d*_{0} denote the initial amount of drug in the blood.

i. Find a difference equation that represents this situation.

ii. Find the amount of drug after 12 hours.

iii. Iterate the difference equation obtained in part I with the initial condition to find the ordered pairs (*n*, *d*_{n}), *n* = 1, 2, … , 28. Graph it.

iv. Find an analytical solution of the obtained difference equation. Use this solution to find the amount of drug in the blood after 1 day.

v. When will the amount of the drug reach 1 mg?

*Discussion*

i. The amount of drug in the blood after (*n* + 1) 4-h periods, *d*_{n}_{+1}, equals the amount of drug after *n* 4-h periods, *d*_{n}, minus 20% of *d*_{n}. We obtain

Therefore this situation ...