Laplace transforms are widely used in engineering, particularly electrical engineering, but there seems to be considerable variation is when they are first encountered in the engineering curriculum. Most of our engineering students take Electric Circuit Theory concurrently with this 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.
Using Laplace transforms to find formulas for solutions frequently involves very tedious arithmetic. This use 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 8.1 and 8.2 form a fairly standard introduction to the definitions and basic techniques for Laplace transforms. Section 8.3 covers delta functions and Section 8.4 is an introduction to the qualitative use of the poles of the Laplace transform.
8.1 Laplace Transforms
The examples of this section are all first-order equations and it can be presented after (or during) Chapter 1, but this is difficult since some of the major rules (and notation for inverse Laplace transforms) don't appear until the next section. The introduction in this section is fairly standard. We have tried to include some motivation for the definition of the Laplace transform and we deal with non-continuous equations earlier than usual. The coefficients in the RC circuit example are not realistic.
Comments on selected exercises
Exercises 1--5, 7 and 17 are computations of Laplace transforms from the definition (filling in the table). Exercises 3, 4, 5, 7 and 17 involve discontinuous functions.
Exercise 6 is computation of two inverse Laplace transforms.
Exercises 8--15 are linear first-order equations, with Exercises 11, 12, 14 and 15 having discontinuities.
Exercises 16 and 18 are derivations of rules for Laplace transform (Laplace transform of an antiderivative and of a rescaling, respectively).
8.2 Laplace Transforms and Second-Order Equations
This section should ideally come after Sections 5.1 and 5.2 since no motivation for considering forced oscillator equations is given here. The remainder of the standard rules and the notation for Inverse Laplace transform is given in this section. The emphasis is on discontinuous forcing terms, otherwise the approach is standard.
Comments on selected exercises
Exercises 1--6 are computations of Laplace transforms from the definition (to fill in the table). Some of these formulas are used later, so if not assignec, these execises should at least be pointed to. The techniques involved are fairly advanced, including differentiating with respect to parameters and induction.
Exercises 7--14 fairly standard forced harmonic oscillator problems which can be done by Laplace transform. Exercises 8, 10, 11 and 14 involve discontinuities. Exercises 9, 10 and 14 involve resonant forcing. Exercise 14 is very complicated.
Exercises 15--19 consider Laplace transforms of periodic forcing functions (square wave and saw tooth) with applications. These exercises require considerably more sophistication than those above.
8.3 Impulse Forcing and Delta Functions
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 8.2, Exercises 15). These are considerably more difficult than Exercises 2--6.
8.4 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 and unforced harmonic oscillator. This is standard in electrical engineering and Figure 8.19 can be found in circuit theory texts. The introduction given here is very brief.
Comments on selected exercises
Exercises 1 and 2 are computations of poles of a Laplace transform.
Exercises 3--6 and 8--11 are familiar forced harmonic oscillator problems. The goal is the use the poles of the Laplace transform to obtain qualitative information about solutions without computing the inverse Laplace transform. Particularly in 5 and 6, analysis of the poles must be combined with common sense, since the forcing term turns off at larger values of $t$. Exercises 10 and 11 refer to square wave and saw tooth forcing (see Section 8.2, Exercises 15--19). Along with Exercise 7, they push the limits of our discussion of poles.
Exercise 7 considers poles on the imaginary axis.
Comments on the Labs
Labs 8.1 and 8.2:
These labs are essentially the same, using delta functions to model the bouncing of a ball. Setting up the equations is very difficult and numerics will be very challenging. A partial answer is quite good for either of these labs.
Return to the Instructor's Manual Table of Contents.
Return to the BU-ODEs home page.
Version 1.1. May, 1996.