Overview of Chapter Six
Laplace transforms are widely used in engineering, particularly electrical engineering. 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). This chapter presents at least the beginning of this theory.
Sections 6.1 and 6.2 form a self-contained introduction to Laplace transforms, where first-order equations with or without discontinuities are discussed. 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.
- DETools (the CD): This is a good place to resurrect HPGSolver for graphical illustrations.
- Diff. Equ. Toolkit (at my web page): Several tools are available to perform specialized calculations to solve some D.E; e.g., Laplace and Inverse Laplace transforms, and partial fractions (Calculus Toolkit), etc., immediately through the World Wide Web.
- Links under Homework (at my web page): Several links are available to help you learn the material; e.g., Table of Laplace transforms and properties, Partial Fractions review and examples with their solutions.
6.1 Laplace Transforms
The basics of Laplace transforms are discussed in this section, with confine applications to first-order equations. Laplace transform of a function f(t) at a particular real s can be seen as a way to s measure “how f(t) is like the function est”: the larger L[f(t)](s), the more f(t) is “like” est.
Comments on selected exercises: 3,9,11,13,15,19,23
Exercises 3,9,11,13 provide practice with the definition of L and L-1.
Exercises 15,19,23 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.
6.2 Discontinuous Functions
This section is a standard presentation of Laplace transforms applied to first-order equations with discontinuous terms. The techniques discussed are mainly algebraic, but you benefit from slope fields just as you did in Chapter 1.
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 diff. Equ.
Comments on selected exercises: 1,3,5,7,9,11,13,17,19
Exercises 1,3 to work with the Heaviside function and to compute Laplace transforms of piecewise-defined functions.
Exercises 5,7 provide practice inverting the transform when it includes terms of the form e-sa.
Exercises 9,11,13; the Laplace transform is used to solve first-order discontinuous initial-value problems.
Exercises 17,19 involve Laplace transforms of periodic forcing functions such as the square wave function. Exercise 17 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. 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: 2,5,7,11,13,15,17,27,29-33.
Exercise 2, 5,7 involve computing the Laplace transforms: directly from the definition, using the fact that cos(w t) satisfies the equation for a simple harmonic oscillator, and a clever way of avoiding the integration.
Exercises 11,13 and 15,17 go together. The first group simply involves completing the square while the second group uses the results to compute inverse Laplace transforms.
Exercises 27,29, 30, 31, 32,33 apply the methods of this section to various initial-value problems.
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: 3,5,7,8,9.
Exercises 3,5 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 8,9 consider periodic delta function forcing (using Section 6.2, Exercise 16).
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: 1,3,6,7,9
Exercises 1,3 involve computing convolutions from the definition.
Exercise 6 is a verification of the commutativity of convolution. It involves the definition of convolution.
Exercises 7,9 reinforce the points made at the end of the section.
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 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 in circuit theory textbooks.
Comments on selected exercises: 1,3, 9
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 Exercise 3, analysis of the poles must be combined with common sense, since the forcing term turns off at larger values of t. Exercise 9 refers to square wave forcing (see Section 6.2, Exercises 17,19).
Comments on the Labs
Lab 6.1: To formulate a conjecture regarding the relationship between multiple poles and growth rates of solutions.
Lab 6.2: To use convolutions to compute the underlying differential equation from a given "experimental data".