Education Technology

Differential Equations

Published on 06/09/2008

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