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Overview of Chapter Six
Using Laplace transforms to find formulas for solutions frequently involves tedious 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 selfcontained 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 firstorder equations with or withoutdiscontinuities. Section 6.3 extends the discussion to secondorderequations. 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. There are no problemsinvolved in skipping 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 firstorder equations.
Comments on selected exercises
Exercises 16 provide practice with the definition of L, andExercises 714 involve L^{1}.
Exercises 1524 are initialvalue 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 firstorderlinear 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 firstorder 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 13, students compute Laplace transforms ofpiecewisedefined functions. They get practice working with theHeaviside function.
Exercises 47 provide practice inverting the transform when itincludes terms of the form e^{sa}.
In Exercises 813, the Laplace transform is used to solve firstorderdiscontinuous initialvalue problems.
Exercises 14 and 15 are also initialvalue problems, but thecomputations are more difficult.
Exercises 1620 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 SecondOrder Equations
This section is a relatively standard discussion of the Laplacetransform method applied to secondorder 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 secondorder forced equations.
Comments on selected exercises
Exercises 14 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 610 also involve computing Laplace transforms, but theinstructions suggest a clever way of avoiding the integration.
Exercises 1114 and 1518 go together. The first group simplyinvolves completing the square while the second group uses the resultsto compute inverse Laplace transforms.
Exercises 1926 show how to use complex arithmetic in place ofcompleting the square. Exercises 2326 are the same asExercises 1518.
Exercises 2733 apply the methods of this section to variousinitialvalue problems. Exercises 2729 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 initialvalueproblems in Exercises 30, 32, and 33.
Exercise 34 suggests another way to do the Laplace transforms inExercises 610.
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.
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 26 are standard secondorder initialvalue problemswith delta function forcing.
Exercise 7 considers the relationship between the delta function and the Heaviside function.
Exercises 810 consider periodic delta function forcing (using Section 6.2, Exercise 16). These problemsare considerably more difficult than Exercises 26.
6.5 Convolution
This is a typical section on convolution. However, it ends with adiscussion of how one can find the solution of an initialvalueproblem without ever knowing the differential equation.
Comments on selected exercises
Exercises 15 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 711 reinforce the points made at the end of thesection.
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 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 larger values of t. Exercises 9 and 10 refer to square wave and sawtooth forcing (see Section 6.2, Exercises 1720).
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.