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.
Follow the Activity procedures:
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