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Overview of Chapter Six
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 6.2form a self-contained introduction to Laplacetransforms. In fact, since some schools would prefer an early introductionto the topic, we have written these sections so that they can becovered immediately after Chapter 1. In other words, we confine ourdiscussion to first-order equations with or withoutdiscontinuities. Section 6.3 extends the discussion to second-orderequations. 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. Youcan easily skip from Section 4.3 directly to Chapter 6.
6.1 Laplace Transforms
The basics of Laplace transforms are discussed in this section. Asmentioned earlier, we confine our discussion to first-order equations.
Comments on selected exercises
Exercises 1-6 provide practice with the definition of L, andExercises 7-14 involve L-1.
Exercises 15-24 are initial-value problems that could have beensolved using integrating factors or the material in Appendix A, butthe problems are written so that they will be solved using Laplacetransforms.
Exercise 25 requests that the students use Laplacetransforms to derive the general solution of a first-orderlinear 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 appliedto first-order equations with discontinuous terms. Even though thetechniques discussed here are mainly algebraic, the students benefitfrom 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 solveris the key to entering discontinuous differential equations.
Comments on selected exercises
In Exercises 1-3, students compute Laplace transforms ofpiecewise-defined functions. They get practice working with theHeaviside function.
Exercises 4-7 provide practice inverting the transform when itincludes terms of the form e-sa.
In Exercises 8-13, the Laplace transform is used to solve first-orderdiscontinuous initial-value problems.
Exercises 14 and 15 are also initial-value problems, but thecomputations are more difficult.
Exercises 16-20 involve Laplace transforms of periodic forcing functions such as the square wave and sawtooth wavefunction. 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 Laplacetransform method applied to second-order linear equations. The most difficult equations considered are those with discontinuousforcing and resonance. The Laplace transform of t sin(w t) isderived in two different ways in the exercise set.This section depends on Sections 4.2 and 4.3 since no motivation isgiven here for considering second-order forced equations.
Comments on selected exercises
Exercises 1-4 involve computing the Laplace transform directly fromthe definition whereas Exercise 5 involves computing the Laplacetransform of cos(w t) using the fact that cos(w t) satisfies theequation for a simple harmonic oscillator.
Exercises 6-10 also involve computing Laplace transforms, but theinstructions suggest a clever way of avoiding the integration.
Exercises 11-14 and 15-18 go together. The first group simplyinvolves completing the square while the second group uses the resultsto compute inverse Laplace transforms.
Exercises 19-26 show how to use complex arithmetic in place ofcompleting the square. Exercises 23-26 are the same asExercises 15-18.
Exercises 27-34 apply the methods of this section to variousinitial-value problems. Exercises 27-29 and 31 could easily be doneusing the methods of Chapter 4, but they are here to give the studentssome practice before they tackle the more complicated initial-valueproblems in Exercises 30 and 32-34.
Exercise 35 suggests another way to do the Laplace transforms inExercises 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 andpounding it on the desk. It also seems to improve their attentionspans.
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 problemswith 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 problemsare considerably more difficult than Exercises 2-6.
6.5 Convolution
This is a typical section on convolution. However, it ends with adiscussion of how one can find the solution of an initial-valueproblem without ever knowing the differential equation.
Comments on selected exercises
Exercises 1-5 involve computing convolutions from thedefinition. Exercise 5 requires a number of trigonometric identities.
Exercise 6 is a verification of the commutativity of convolution. Itinvolves the definition of convolution.
Exercises 7-11 reinforce the points made at the end of thesection.
6.6 The Qualitative Theory of Laplace Transforms
This sectionis 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 playthe same role as the eigenvalues for an unforced harmonic oscillator. This point of viewis standard in electrical engineering, and Figure 6.26can be found circuit theory textbooks.
Comments on selected exercises
All of these exercises involve familiar equations that model forced harmonicoscillators. 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 answersare (usually) one or two sentences. Most (but not all) arerelatively straightforward.
Exercise 11 is a "multiple choice" matching exercise. Thestudents match functions with their transforms from a listof transforms that is larger than the list of functions.
Exercises 12-17 are true/false problems. We always expectour students to justify their answers.
Exercises 18-23 give students practice computing inverseLaplace transforms.
In Exercises 24 and 25, the student is asked to solve theequation in two ways---using the methods of Chapter 4and using the Laplace transform. Then they are asked tocomment on which method they prefer.
Exercises 26-30 provide practice solvinginitial-value problems that involve Heaviside functions anddelta functions.
Comments on the Labs
Lab 6.1: Poles
In this lab, the students are asked to formulate a conjectureregarding the relationship between multiple poles and growth rates ofsolutions.
Lab 6.2: Convolutions
In this lab, the students are given "experimental data" and areasked to use convolutions to compute the underlying differentialequation.