|Module 20 - Differential Equations and Euler's Method|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test|
|Lesson 20.1: Logistic Growth Revisited|
Module 3 contained an experiment that simulated the spread of a disease through a small population. In this lesson you will revisit the data used to model the spread of a disease and solve the related differential equation.
Modeling the Data
The following data from a simulation gives the cumulative number of infected people as a disease spreads through a population of 25 people.
The shape of this scatter plot is called a logistic curve. The differential equation that describes the rate at which the disease spreads is
y' = k * y * (M y)
where y is the number of infected people, t is time measured in days, M is the maximum possible number of infected people 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 equations for different values of k . Use an initial guess of 0.05 for k.
Remember to include the multiplication sign before the parenthesis.
20.1.1 Modify k until the graph of the solution to the differential equation approximates the scatter plot.
After you have found the value of k, use deSolve to find the solution to the differential equation.
This graph was generated in the answer to 20.1.1 using the numerical Differential Equation solver in Differential Equation graphing mode that does not result in a formula for the solution. This time we generated the graph by solving the Differential Equation symbolically and graphed the solution in Function Mode.
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