Since the entire chapter deals almost exclusively with first-order equations, most of this material can be covered immediately after Section 1.4 (Euler's method). However, Sections 7.1 and 7.4 are somewhat more sophisticated than the rest of the text, so it may be appropriate to skim Section 7.1 and jump directly to Section 7.2.

We expect that the students will use some type of computer technology as they solve most of the exercises in this chapter.

DETools

Both EulersMethod and EulersMethodForSystems provide four different step sizes for Euler's method as well as Runge-Kutta with one step size. These tools may help illustrate some of the ideas discussed in this chapter, but alone they are probably not sufficient. Actually it is surprisingly easy to do most of the necessary computation using a spreadsheet.

7.1 Numerical Error in Euler's Method

This section is a detailed discussion of the errors involved in Euler's method including an explanation of the fact that Euler's method is a first-order method. Precise error bounds are derived in Exercises 11 and 12, but the text contains a discussion of the reasons why estimates of this type are often far too conservative.

Comments on selected exercises

Exercises 1-5 study the errors involved in Euler's method as a function of step size.

Exercise 6 is a good exercise to see if the students have a practical understanding of the fact that Euler's method is a first-order method.

Exercises 7-9 use a convenient method to estimate the error in an approximation.

Exercises 10-12 are theoretical. In particular, Exercise 11 is a long step-by-step derivation of precise error bounds.

7.2 Improving Euler's Method

Improved Euler's method is introduced as an example of a second-order method. One goal of the section is to develop the analogy between numerical methods to approximate solutions to differential equations and numerical methods to approximate integrals. The other goal is to contrast the accuracy of a second-order method with that of a first-order method (Euler's method).

Comments on selected exercises

Exercises 1-4 of this section involve the initial-value problems in Exercises 1-4 of Section 1.4. Similarly, Exercises 5-8 involve the initial-value problems in Exercises 6-9 of Section 1.4. These exercises compare the accuracy of improved Euler's method with that of Euler's method. The computations are possible by hand, but they would be tedious.

Exercises 9-13 give students practical experience with the fact that improved Euler's method is a second-order method.

Exercises 14-17 compare the accuracy of Euler's method versus improved Euler's method.

7.3 The Runge-Kutta Method

In this section the (fourth-order) Runge-Kutta algorithm is derived. As this is a method of choice for many applications, implementations of the method are given for the TI calculator, in Mathematica , and in the C programming language. We also use vector notation to discuss Runge-Kutta for first-order systems.

Comments on the exercises

In Exercises 1-5, we revisit initial-value problems that were studied in Sections 1.4 and 7.2. Using the results, we can compare the accuracy of Runge-Kutta with that of Euler's and improved Euler's methods.

Exercise 6 gives students practice with the fact that Runge-Kutta is a fourth-order method.

In Exercises 7 and 8, we illustrate how Runge-Kutta can be used to approximate solutions to first-order systems as well as second-order equations.

7.4 The Effects of Finite Arithmetic

The goal of this short section is to point out that, in practice, using a ridiculously small step size may not yield more accurate numerical approximations. The exercises study this phenomenon for particular examples.

Review Exercises

Exercises 1-5 are true/false problems. We always expect our students to justify their answers.

Exercise 6 compares the results of Euler's method, improved Euler's method, and Runge-Kutta for a first-order equation.

Exercise 7 compares the results of Euler's method, improved Euler's method, and Runge-Kutta for a competing species system.

Comments on the Labs

Lab 7.1

This lab relates the geometry of an equation with the accuracy of the numerical approximation. It addresses the question of why some methods are more accurate than expected on certain equations.

Lab 7.2

In this lab, numerical methods are used to approximate solutions of Newton's equations. Parts of the lab could be used as detailed exercises or projects much earlier in the text.