Using Laplace transforms to find formulas for solutions frequently involves a lot of algebra. This aspect of Laplace transforms will disappear as symbolic software becomes cheaper and better. However, Laplace transforms can be used as a tool for qualitative analysis of equations. The poles of the Laplace transform fill the same role as the eigenvalues of a linear system (of which they are a generalization). We have attempted to present at least the beginning of this theory.

Sections 6.1 and 6.2 form a self-contained introduction to Laplace transforms. In fact, since some schools would prefer an early introduction to the topic, we have written these sections so that they can be covered immediately after Chapter 1. In other words, we confine our discussion to first-order equations with or without discontinuities. Section 6.3 extends the discussion to second-order equations. Delta functions are covered in Section 6.4, and convolution is discussed in Section 6.5. Section 6.6 is an introduction to the qualitative use of the poles of the Laplace transform.

This chapter is independent of Chapter 5. You can easily skip from Section 4.3 directly to Chapter 6.

6.1 Laplace Transforms

The basics of Laplace transforms are discussed in this section. As mentioned earlier, we confine our discussion to first-order equations.

Comments on selected exercises

Exercises 1-6 provide practice with the definition of L, and Exercises 7-14 involve L^-1.

Exercises 15-24 are initial-value problems that could have been solved using integrating factors or the material in Appendix A, but the problems are written so that they will be solved using Laplace transforms.

Exercise 25 requests that the students use Laplace transforms to derive the general solution of a first-order linear equation.

Exercise 27 illustrates why the equations considered in this chapter are always linear.

6.2 Discontinuous Functions

This section is a standard presentation of Laplace transforms applied to first-order equations with discontinuous terms. Even though the techniques discussed here are mainly algebraic, the students benefit from slope fields just as they did in Chapter 1.

DETools

This is a good place to resurrect HPGSolver. It really helps with the explanation of the terms in the solution that involve the Heaviside function. Use of the step function in the solver is the key to entering discontinuous differential equations.

Comments on selected exercises

In Exercises 1-3, students compute Laplace transforms of piecewise-defined functions. They get practice working with the Heaviside function.

Exercises 4-7 provide practice inverting the transform when it includes terms of the form e^-sa.

In Exercises 8-13, the Laplace transform is used to solve first-order discontinuous initial-value problems.

Exercises 14 and 15 are also initial-value problems, but the computations are more difficult.

Exercises 16-20 involve Laplace transforms of periodic forcing functions such as the square wave and sawtooth wave function. Exercises 17 and 18 can be done without using Exercise 16, but Exercise 16 simplifies the calculation considerably.

6.3 Second-Order Equations

This section is a relatively standard discussion of the Laplace transform method applied to second-order linear equations. The most difficult equations considered are those with discontinuous forcing and resonance. The Laplace transform of t sin(w t) is derived in two different ways in the exercise set. This section depends on Sections 4.2 and 4.3 since no motivation is given here for considering second-order forced equations.

Comments on selected exercises

Exercises 1-4 involve computing the Laplace transform directly from the definition whereas Exercise 5 involves computing the Laplace transform of cos(w t) using the fact that cos(w t) satisfies the equation for a simple harmonic oscillator.

Exercises 6-10 also involve computing Laplace transforms, but the instructions suggest a clever way of avoiding the integration.

Exercises 11-14 and 15-18 go together. The first group simply involves completing the square while the second group uses the results to compute inverse Laplace transforms.

Exercises 19-26 show how to use complex arithmetic in place of completing the square. Exercises 23-26 are the same as Exercises 15-18.

Exercises 27-34 apply the methods of this section to various initial-value problems. Exercises 27-29 and 31 could easily be done using the methods of Chapter 4, but they are here to give the students some practice before they tackle the more complicated initial-value problems in Exercises 30 and 32-34.

Exercise 35 suggests another way to do the Laplace transforms in Exercises 6-10.

6.4 Delta Functions and Impulse Forcing

This is a relatively standard section on the Dirac delta function. The "limit" approach is used. Thinking of the delta function as the "derivative" of the Heaviside function is discussed in Exercise 7.

Our students seem to enjoy our bringing a hammer to class and pounding it on the desk. It also seems to improve their attention spans.

Comments on selected exercises

In Exercise 1, the limit required to compute the Laplace transform of the delta function is computed (L'Hôpital's rule).

Exercises 2-6 are standard second-order initial-value problems with delta function forcing.

Exercise 7 considers the relationship between the delta function and the Heaviside function.

Exercises 8-10 consider periodic delta function forcing (using Section 6.2, Exercise 16). These problems are considerably more difficult than Exercises 2-6.

6.5 Convolution

This is a typical section on convolution. However, it ends with a discussion of how one can find the solution of an initial-value problem without ever knowing the differential equation.

Comments on selected exercises

Exercises 1-5 involve computing convolutions from the definition. Exercise 5 requires a number of trigonometric identities.

Exercise 6 is a verification of the commutativity of convolution. It involves the definition of convolution.

Exercises 7-11 reinforce the points made at the end of the section.

6.6 The Qualitative Theory of Laplace Transforms

This section is not a standard one. It is a brief introduction to how Laplace transforms can be used to obtain qualitative information. The emphasis is on the idea that the poles of a Laplace transform of a solution for a forced harmonic oscillator play the same role as the eigenvalues for an unforced harmonic oscillator. This point of view is standard in electrical engineering, and Figure 6.26 can be found circuit theory textbooks.

Comments on selected exercises

All of these exercises involve familiar equations that model forced harmonic oscillators. The goal here is the use the poles of the Laplace transform to obtain qualitative information about solutions without computing the inverse Laplace transform. Particularly in Exercises 3 and 4, analysis of the poles must be combined with common sense, since the forcing term turns off at some value of t. Exercises 9 and 10 refer to square wave and sawtooth forcing (see Section 6.2, Exercises 17-20).

Review Exercises

Exercises 1-10 are "short answer" exercises. The answers are (usually) one or two sentences. Most (but not all) are relatively straightforward.

Exercise 11 is a "multiple choice" matching exercise. The students match functions with their transforms from a list of transforms that is larger than the list of functions.

Exercises 12-17 are true/false problems. We always expect our students to justify their answers.

Exercises 18-23 give students practice computing inverse Laplace transforms.

In Exercises 24 and 25, the student is asked to solve the equation in two ways---using the methods of Chapter 4 and using the Laplace transform. Then they are asked to comment on which method they prefer.

Exercises 26-30 provide practice solving initial-value problems that involve Heaviside functions and delta functions.

Comments on the Labs

Lab 6.1: Poles

In this lab, the students are asked to formulate a conjecture regarding the relationship between multiple poles and growth rates of solutions.

Lab 6.2: Convolutions

In this lab, the students are given "experimental data" and are asked to use convolutions to compute the underlying differential equation.