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Ch6 The Laplace Transform

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Ordinary Differential Equations (Math 2302)  Õæ Aë áÖß @ .X . @ A®‚Ë@ 2017-2016 úGAJË@ ɒ®Ë@ Chapter 6 The Laplace Transform Many practical engineering problems involve mechanical or electrical systems acted on by discontinuous or impulsive forcing terms. For such problems the methods described in Chapter 3 are often rather awkward to use. Another method that is especially well suited to these problems, although useful much more generally, is based on the Laplace transform. The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform (now called the z−transform) in his work on probability theory. However, the techniques described in this chapter were not developed until a century or more later. They are due mainly to Oliver Heaviside (1850–1925), an innovative but unconventional English electrical engineer, who made significant contributions to the development and application of electromagnetic theory. The Laplace transforms are simple so they are frequently used to solve a wide class of linear differential equations. Like other transforms, Laplace transforms are used to determine particular solutions. 6.1 Definition of the Laplace Transform An integral transform is a relation of the form Z β K(s, t)f (t)dt, F (s) = α where a given function f is transformed into another function F by means of an integral. The function F is called the transform of f , and the function K is called the kernel of ...
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