§7.5 Modeling with differential equations

Introduction

Developing a differential equation

Any time that the rate of change of a quantity is related to the amount of a quantity, a differential equation naturally arises. In the following prompts, we encouner two such scenarios; for each, we want to develop a differential equation whose solution is the quantity of interest.

Suppose you have a bank account in which money grows at an annual rate of 3%.

If you have ＄10,000 in the account, at what rate is your money growing?

In the Great Lakes region, rivers flowing into the lakes carry a great deal of pollution in the form of small pieces of plastic averaging 1 millimeter in diameter. In order to understand how the amount of plastic in Lake Michigan is changing, construct a model for how this type of pollution has built up in the lake.

What is the rate of change of $A$ with respect to $t$?

Suppose that you are also withdrawing ＄10,000 per year. Write a differential equation that expresses the total rate of change of $A$.

Sketch a slope field for this differential equation, find any equilibrium solutions, and identify them as either stable or unstable. Write a sentence or two that describes the significance of the stability of the equilibrium solution.

Suppose that you initially deposit ＄100,000 into the account. How long does it take for you to deplete the account?

What is the smallest amount of money you would need to have in the account to guarantee that you never deplete the money in the account?

If your initial deposit is ＄300,000, how much could you withdraw every year without depleting the account?

Write a differential equation for $M(t)$, the amount of morphine in the patient's bloodstream, using $k$ as the constant proportionality.

Assuming that the initial dose of morphine is $M_0$, solve the initial value problem to find $M(t)$. Use the fact that the half-life for the absorption of morphine is two hours to find the constant $k$.

Suppose that a patient is given morphine intravenously at the rate of 3 milligrams per hour. Write a differential equation that combines the intravenous administration of morphine with the body's natural absorption.

Find any equilibrium solutions and determine their stability.

Assuming that there is initially no morphine in the patient's bloodstream, solve the initial value problem to determine $M(t)$. What happens to $M(t)$ after a very long time?

To what rate should a doctor reduce the intravenous rate so that there is eventually 7 milligrams of morphine in the patient's bloodstream?