Chapter 30

Laplace Transform Methods

30.1 Linear O.D.E. With Constant CoefficientsThis method of Laplace1 transforms succeeds so well because of the algebraic technique ofpartial fractions and the fact that the Laplace transform is a linear mapping. It works verywell to solve higher order initial value problems involving linear equations with constantcoefficients and also more generally first order systems. It is all about changing a differen-tial equation into an algebraic equation, solving that one, and then extracting the solutionto the original differential equation from what was obtained.

This presentation will emphasize the algebraic procedures. The analytical questions arenot trivial and are given a discussion in Section 4.2.

For an initial value problem, you can often reduce to one which has initial conditiongiven at 0 by simply changing the independent variable. Also, this is where the initialcondition is typically given anyway so in this method, I will assume all the initial conditionsare given at 0.

Definition 30.1.1 Let f be a function defined on [0,∞) which has exponential growth,meaning that

| f (t)| ≤Ceλ t

for some real λ . Then the Laplace transform of f , denoted by L ( f ) is defined as

F (s)≡L f (s) =∫

∞

0e−ts f (t)dt

for all s sufficiently large. It is customary to write this transform as F (s) or L f (s) andthe function as f (t) instead of f . In other words, t is considered a generic variable as is sand you tell the difference by whether it is t or s. It is sloppy but convenient notation.

Lemma 30.1.2 L is a linear mapping in the sense that if f ,g have exponential growth,then for all s large enough and a,b scalars,

L (a f (t)+bg(t))(s) = aL f (s)+bL g(s)1Pierre-Simon, marquis de Laplace (1749-1827) had interests in mathematics, physics, probability, and as-

tronomy. He wrote a major book called celestial mechanics. There is also the Laplacian named after him, andLaplace’s equation in potential theory. The expansion of a determinant along a row or column is called Laplaceexpansion. He was also involved in the development of the metric system. It is hard to overstate the importanceof his contributions to mathematics and the other subjects which interested him. [14]

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