# Activities

• • • ##### Subject Area

• Math: Calculus: Applications of the Derivative

• ##### Author College

60 Minutes

• TI-86
• ##### Other Materials
This is Activity 8 from the EXPLORATIONS Book: Using the TI-86 in Collegiate Mathematics: A Tutorial

## Differential Equations

#### 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.

• 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