Activity Overview
In this activity, students analyze first order, second order, and autonomous systems of two first order differential equations.
Before the Activity
Set up the calculator as explained in the activity
See the attached PDF file for detailed instructions for this activity
Print pages 103 - 120 from the attached PDF file for the class
During the Activity
Distribute the pages to the class.
Follow the Activity procedures:
Enter a differential equation and draw a slope field for it
Use the Runge-Kutta (RK) algorithm to calculate and graphically display an approximate solution for a first order initial value problem
Graph the solution of the initial value problem over two halves of the interval, and then paste the pieces together to obtain the graph over the entire interval
Study Euler's method to approximate a solution, compare it with the Runge Kutta method, and realize that the RK method is vastly superior
Understand the Lotka-Volterra system, used to model a predator-prey environmental situation
Enter a system of two first order equations, set up the algorithm, and graph the solutions
Recognize the fact that the solutions are periodic and have the same period, which is determined using the Trace feature
Further analyze the Lotka-Volterra system, and graph the trajectory for the system in the phase-plane
Graph a direction field for the autonomous system to visualize its trajectories
Use the Explore feature to graph the trajectories passing through different points
Use the built-in algorithm to calculate and graph the solution of a second order initial value problem over two halves of the interval, and then paste the pieces together to obtain the graph over the entire interval
After the Activity
Students complete the exercises on the Activity sheet.
Review student results:
As a class, discuss questions that appeared to be more challenging
Re-teach concepts as necessary