# Chapte 4 - CHAPTER 4 The Laplace Transform 4.1 Introduction...

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CHAPTER 4 The Laplace Transform 4.1 Introduction The Laplace transform provides an eﬀective method of solving initial-value problems for linear diﬀerential equations with constant coeﬃcients. However, the usefulness of Laplace transforms is by no means restricted to this class of problems. Some understanding of the basic theory is an essential part of the mathematical background of engineers, scientists and mathematicians. The Laplace transform is deFned in terms of an integral over the interval [0 , ). In- tegrals over an inFnite interval are called improper integrals, a topic studied in Calculus II. DEFINITION Let f be a continuous function on [0 , ). The Laplace transform of f , denoted by L [( f ( x )], or by F ( s ), is the function given by L [ f ( x )] = F ( s )= ± 0 e - sx f ( x ) dx. (1) The domain of F is the set of all real numbers s for which the improper integral converges. In more advanced treatments of the Laplace transform the parameter s assumes com- plex values, but the restriction to real values is suﬃcient for our purposes here. Note that L transforms a function f = f ( x ) into a function F = F ( s ) of the parameter s . The continuity assumption on f will hold throughout the Frst three sections. It is made for convenience in presenting the basic properties of L and for applying the Laplace transform method to solving initial-value problems. In the last two sections of this chapter we extend the deFnition of L to a larger class of functions, the piecewise continuous functions on [0 , ). There we will apply L to the problem of solving nonhomogeneous equations in which the nonhomogeneous term is piecewise continuous. This will involve some extension of our concepts of diﬀerential equation and solution. As indicated above, the primary application of Laplace transforms of interest to us is solving linear diﬀerential equations with constant coeﬃcients. Referring to our work in Chapter 3, the functions which arise naturally in the treatment of these equations are: p ( x ) e rx ,p ( x ) cos βx, p ( x ) sin p ( x ) e cos p ( x ) e sin βx where p is a polynomial. We begin by calculating the Laplace transforms of some simple cases of these functions. 115

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Example 1. Let f ( x )=1 · e 0 x 1o n[ 0 , ). By the Defnition, L [1] = ± 0 e - sx · 1 dx = lim b →∞ ± b 0 e - sx dx = lim b →∞ ² e - sx - s ³ ³ ³ ³ b 0 ´ = lim b →∞ µ e - sb - s
The following table gives a basic list of the Laplace transforms of functions that we will encounter in this chapter. While the entries in the table can be veriFed using the DeFnition, some of the integrations involved are complicated. The properties of the Laplace transform presented in the next section provide a more eﬃcient way to obtain many of the entries in the table. Handbooks of mathematical functions, for example the CRC Standard Mathematical Tables, give extensive tables of Laplace transforms.

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Chapte 4 - CHAPTER 4 The Laplace Transform 4.1 Introduction...

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