Overview of Chapter Six

Laplace transforms are widely used in engineering, particularly electrical engineering, but there seems to be considerable variation in when they are first encountered in the engineering curriculum. Most of our engineering students take electric circuit theory concurrently with our course. However, Laplace transforms are not used until the signals and systems course the subsequent semester, and that course is not taken by all engineering students at Boston University.

Using Laplace transforms to find formulas for solutions frequently involves tedious arithmetic. 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 entends the discussion to second-order equations. Delta functions are covered in Section 6.4, convolution is covered in Section 6.5, and Section 6.6 is an introduction to the qualitative use of the poles of the Laplace transform.

6.1 Laplace Transforms

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

Comments on selected exercises

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

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 similar to those that are used in Chapter 1.

Comments on selected exercises

Exercises 1--13 are standard for this section.

Exercises 16--20 consider Laplace transforms of periodic forcing functions (square wave and saw tooth) with applications. These exercises require considerably more sophistication than Exercises 1--13.

6.3 Second-Order Equations

This section should come after Chapter 4 since no motivation for considering equations for forced oscillators is given here. The techniques of Section 6.1 are extended so that second-order equations can be studied.

Comments on selected exercises

Exercises 1--32 are standard for this section. Exercises 29--36 also use the techniques developed in Section 6.2.

6.3 Delta Functions and Impulse Forcing

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

Comments on selected exercises

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

Exercises 2--6 are standard forced harmonic oscillators 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 brief section on convolution which concludes 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--4 are routine, but the remaining exercises in this section are quite a bit more challenging.

6.6 The Qualitative Theory of Laplace Transforms

This is the only nonstandard section in this chapter, and it is only 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 fill the same role as the eigenvalues for an unforced harmonic oscillator. This is standard in electrical engineering and Figure 6.26 can be found in circuit theory texts.

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 larger values of t. Exercises 9 and 10 refer to square wave and saw tooth forcing (see Section 6.2, Exercises 17--20).

Comments on the Labs

Lab 6.1: Convolutions

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

Lab 6.2: Poles

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