# Activities

• • • ##### Subject Area

• Math: Calculus: Other Functions

• ##### Author 9-12

90 Minutes

• TI-86
• ##### Other Materials
This is Activity 8 from the EXPLORATIONS Book:
Differential Equations With The TI-86

## Numerical Methods and Series

#### Activity Overview

In this activity, students take a closer look at numerical methods used to solve initial-value differential equations, including the methods used internally by the calculator. They also learn to solve numerical computations for the equations where the calculator can not be used.

#### Before the Activity

• See the attached PDF file for detailed instructions for this activity
• Print pages 77 - 94 from the attached PDF file for your class
• #### During the Activity

Distribute the pages to the class.

Graphical Euler's Method:

• Enter the differential equation
• Change the values of the number of Euler steps, number of steps between plotted points, change in t for plotted points, and graph

• Numerical Euler's method:
• Enter the differential equation and select Euler format
• Use six different values of h the stepsize
• Confirm the expected relationship between h and size of the final t-value
• Estimate the number of Euler steps needed to have a final accuracy of 1E-7

• Solution Series:
• Use differentiation to get higher derivatives
• Solve the equation using Taylor series expansion, knowing the derivative at t = 0
• Note the method of using Taylor series involves too much symbolic work while the Runge-Kutta method is simpler
• Use the Runge-Kutta method to solve the differential equation

• Numerical Runge-Kutta Method
• Write a small program to implement the classical third-order Runge-Kutta method to solve the differential equation, and also include lists of the local error (for the first step) and global error for variety of stepsizes
• Vary the values of stepsize and total number of Euler steps to get a more accurate outcome

• Numerical internal RK Method
• Re-enter the equation and change the parameters and initial conditions
• Repeatedly change the difTol and explore the effect on the global error, and time

• Graphical internal RK Methods
• Enter the system of equations and graph them with separate styles
• Use the difTol function to set the parameters
• Use eval command to compare the final computed values for the different tolerances
• #### After the Activity

Students complete the problems on the exercise page.

Review student results:

• As a class, discuss questions that appeared to be more challenging
• Re-teach concepts as necessary