CSC: Modeling the USA Population

Chapter 3: Solving the Differential Equations

Abstract : In this chapter, we use integration to solve the three differential equations that arose in Chapter 2. We also study Euler's method as a technique for obtaining numerical (i.e., approximate) solutions of differential equations. For comparison with the analytical results of the past chapters, we use Euler's method to fit curves to the USA population census data for the periods 1790-1850 and 1950-1990. In the latter case, we have to make an assumption about the limiting value of the size of the population.

Prerequisite Knowledge : A conceptual introduction to integrals; integrals of polynomial functions; integral of .

In this chapter we will be using our knowledge of solving differential equations by integration to continue our efforts to model the USA population from census data. There are three differential equations that commonly arise in the study of population models. They are:

, where is a constant.

, where and are constants.

, where and are constants, .

We have met the first two equations in Chapter 2, and we will use the third in Chapter 4. You should explain in your discussion below why the first equation states that the relative growth rate is constant and the second that it is a linear function of . The third equation may look a little strange, but it is just as important as the first two. It is called the logistic equation. When applied to populations, it is also known as the Verhulst model in honor of the Belgian mathematician who proposed it in 1838. After we have solved this equation, you should be able to explain in your discussion why it is a model of limited growth and the relevance of the constant in this regard.

In this chapter, we will solve all three differential equations analytically using the method of separation of variables, and we will obtain approximate solutions numerically using Euler's method.

Analytic Methods

Slope Fields

Euler's Method

Studying the Period 1790-1850 with Euler's Method

Studying the Period 1950-1990 with Euler's Method