Module 22 - Differential Equations and Euler's Method

Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test

Lesson 22.1: Logistic Growth Revisited

Module 5 contained an experiment that simulated the spread of a rumor through a small population. In this lesson you will revisit that data and solve the related differential equation.

Modeling the Data

The following data from a simulation gives the cumulative number of people who know a rumor as it spreads through a population of 25 people.

 Day Number Who Knew 1 1 2 2 3 4 4 8 5 13 6 20 7 24 8 25

• Enter the data in your Stats/List Editor, set up Plot1, and create a scatter plot by pressing and selecting 9:ZoomStat.

[0.3, 8.7, 1] x [-3.08, 29.08, 10]

The shape of this scatter plot is called a logistic curve. The differential equation that describes the rate at which the rumor spreads is

y ' = k · y · (My)

where y is the number of people who know the rumor at time t. t is time measured in days, M is the maximum possible number of people who can know the rumor, and k is a constant. For this simulation, M is 25.

You will use the "Guess and Check" method to find an approximation for k that fits the data by graphing solutions to the differential equation for different values of k. Use an initial guess of 0.05 for k.

• Set Y1 = 0.05Y*(25-Y).