# 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

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.

*n*,

*d*),

_{n}*n*= 1, 2, … , 28. Graph it.

*Discussion*

*n*+ 1) 4-h periods,

*d*

_{n}_{+1}, equals the amount of drug after

*n*4-h periods,

*d*, minus 20% of

_{n}*d*. We obtain

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