IntroLaplaceTransform

IntroLaplaceTransform - The Laplace Transform Among the...

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The Laplace Transform Among the tools are useful for solving differential linear equations are integral transforms. Definition : An integral transform is a relation of the form F(s) = K(s,t)f(t)dt a b ! where K(s,t) is a given function of the variables s and t, called the kernel of the transformation and the limits of integration a and b are given and it is possible that a = - and b = . The relation transforms the function f(t) into another function F(s), which is called the transform of f. The general idea of using an integral transform to solve a linear differential equation is to transform a problem for an unknown function f into a simpler problem for the function F. Then, solve the simpler problem to find F and then recover the function f from its transform F. This last step is known as inverting the transform. There are several integral transform that are useful to solve differential equation, but in this chapter we will use the Laplace transform. Definition : Let f(t) be given for t 0 and assume the function satisfy certain conditions to be stated later on. The Laplace transform of f(t), that it is denoted by L {f(t)} or F(s), is defined by the equation L {f(t)} = F(s) = e ! st f(t)dt 0 " # whenever the improper integral converges. Remark
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IntroLaplaceTransform - The Laplace Transform Among the...

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