# 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 dn denote the amount of drug in the blood after n 4-h periods, and d0 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, dn), 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, dn+1, equals the amount of drug after n 4-h periods, dn, minus 20% of dn. We obtain
Therefore this situation ...

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