Teaching the Laplace Transform Using DiagramsAuthor(s): V. Ngo and S. OuzomgiSource: The College Mathematics Journal, Vol. 23, No. 4 (Sep., 1992), pp. 309-312Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2686950 .Accessed: 07/08/2013 17:04

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I claim that the simplifying assumption here is made too early in the procedure, for to most students it is completely unmotivated at this point. In fact, often, the only reason given for the simplifying assumption is that it reduces the amount of work required in determining yp(x); unfortunately, this leaves many students wondering why none of the other terms was chosen to be zero instead, for that would just as well make evaluating y"p(x) easier.

I propose that, rather than making the simplifying assumption at this stage, it is far better to labor through evaluating yp(x) in all its goriness. For doing so gives (after some rearrangement)

f=y'p+py'p + 0) is defined by

F(s)=^{f(t)} = [ e-?f(t)dt

for all values s for which the integral is defined. We write f(t)=^f l{F(s)} and call f(t) the inverse Laplace transform of F(s). If a is a real constant and n is a

VOL. 23, NO. 4, SEPTEMBER 1992 309

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positive integer, we have

1 ^{eat} s ? a

a

(s>a) sin n\

^flsin at) = ~ ~ 1 J s2 + a2 (s>0) ^f{cos at) =

s2 + a2

(s>0)

(s>0).

The following theorem can be found in standard differential equations texts.

Theorem 1. Let f(t) be a function with Laplace transform F(s). Then the Laplace transform of the function f^f^dr is (l/s)F(s).

This theorem is illustrated by the following diagram.

/(') i-1 H*)

ff{r)di

l -F(s)

This diagram indicates that integrating f(t) corresponds to dividing its Laplace transform by s.

Calculation of Laplace transforms and their inverses may be illustrated in a similar way.

Example. Find the inverse Laplace transform of l/(s2(s - 1)).

Solution. We see that \/(s2(s - 1)) can be obtained from l/(s-\), whose inverse Laplace transform is el, by dividing by s twice. We draw the corresponding diagram, filling in the appropriate inverse transforms. The answer is in the lower left corner.

1

(teTdr = et-l

(\eT-l)dT = ?t-1

s-l

1

s(s-l)

s2(s-l)

Other properties can be illustrated in the same manner.

Theorem 2. Let f(t) be a function with Laplace transform F(s). Then the Laplace

transform of the function tf(t) is -F'(s).

f(0

tfO)

F(s)

F'(s)

310 THE COLLEGE MATHEMATICS JOURNAL

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This diagram indicates that multiplying f(t) by t corresponds to differentiating its Laplace transform and multiplying by -1.

Example. Find the inverse Laplace transform f(t) of \n(s2/(s2 + 1)).

Solution. We construct the following diagram

/(o

tf(t) = -2 + 2cos?

ln- V + i)

= 2 In s - \n(s2 + 1)

2s

s s2+\

from which we see that tf(t)= -2 + 2cos t and so /(') = 2cosr-2

t

Two other important properties of the Laplace transform can be illustrated by the self-explanatory diagrams

/(')

eatf(t)

F(s) /(0

F(s-a) ua(t)f(t-a)

F(s)

e~asF(s)

where uit) ?? 1 t > a

jO r

We have found this diagrammatic approach very effective in enforcing the concept of the Laplace transform and its properties, while also providing a visual way of keeping track of the processes used in calculations.

Reference

R. V. Churchill, Operational Mathematics, 3rd ed., McGraw Hill, New York, 1972.

A Serendipitous Application of the Pythagorean Triplets Susan Forman, Bronx Community College/CUNY, Bronx, NY 10453

On occasion, a purely pedagogical consideration leads to an interesting mathemati? cal result. To determine whether students had a good grasp of the process of

factoring monic quadratic polynomials, I asked them to factor pairs of the form

(x2+px + _7, x2+px -q), P,q^0 (1)

where each polynomial has integer zeros. Examples are:

(i) (x2 + 5x + 6, x2jt5x-6)

(ii) (x2 + 13* + 30, x2 + 13* - 30)

(iii) (x2 4- Ux + 60, x2 + Ux - 60).

The natural question arose as to whether it is possible to produce all such pairs of polynomials. As we will see, the answer is yes.

We begin with the observation that once a pair of the desired polynomials is known, an infinite number of pairs can be generated from it; for if the polynomials in (1) have integer zeros, so do (x2 + tpx 4- t2q, x2 4- tpx - t2q) for each integer t. We are merely producing polynomials whose zeros have been multiplied by t. This motivates the following definitions:

Definitions. A pair of polynomials of the form described in (1) is an integer quadratic polynomial pair if each polynomial has integer zeros. If the polynomials in (1) satisfy the additional requirement that (p,q) = 1, then the polynomials will be called a representative pair.

It is readily seen that if t\(p,q) and (x2+px + q, x2+px-q) is an integer quadratic polynomial pair, then (x2 4- (p/t)x 4- q/t2, x2 4- (p/t)x - q/t2) is also such a pair. It follows that each integer quadratic polynomial pair can be derived from a representative pair and therefore, it suffices to produce all representative pairs. To this end, we prove the following theorem.

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Article Contentsp. 309p. 310p. 311p. 312

Issue Table of ContentsThe College Mathematics Journal, Vol. 23, No. 4 (Sep., 1992), pp. i+274-355Front Matter [pp. ]The Joy of Mathematics: A Mary P. Dolciani Lecture [pp. 274-281]Optimal Locations [pp. 282-289]Using Simulation to Study Linear Regression [pp. 290-295]How Should We Introduce Integration? [pp. 296-298]Gems of Exposition in Elementary Linear Algebra [pp. 299-303]Fallacies, Flaws, and Flimflam [pp. 304-307]Classroom CapsulesTiming Is Everything [pp. 308-309]Teaching the Laplace Transform Using Diagrams [pp. 309-312]A Serendipitous Application of the Pythagorean Triplets [pp. 312-314]Erratum: Relative Maxima or Minima for a Function of Two Variables [pp. 314]

Student Research ProjectsA Sliding Block Problem [pp. 315-319]

Computer CornerUsing Fourier Analysis in Digital Signal Processing [pp. 320-328]

Classroom Computer CapsulesInterpolating Polynomials and Their Coordinates Relative to a Basis [pp. 329-333]

Software Reviews [pp. 334-339]Problems and Solutions [pp. 340-347]Media Highlights [pp. 348-352]Book ReviewReview: untitled [pp. 353-355]

Back Matter [pp. ]