Laplace Transform Methods

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CHAPTER 1Laplace Transform MethodsLaplace Transform is a method frequently employed by applying the Laplace Transform , one can change an ordinary dif-ferential equation into an algebraic equation, as algebraic equation isgenerally easier to deal with. Another advantage of Laplace transformis in dealing the external force is either impulsive , (the force lasts avery shot time period such as the bat hits a baseball) or the force is onand off for some regular or irregular period of The Laplace TransformIff(t) is defined over interval [0, ), the Laplace Transform off,denoted as f(s),isL(f) = f(s) = 0e stf(t)dtOur first theorem states when Laplace Transform can be performed, (t)is (piecewise) continuous and there are pos-itive numbersM, asuch that|f(t)| Meatfor allt cThen f(s)is defined for alls > cThe next result shows that Laplace Transform is unique in the sensethat different continuous functions will have different Laplace f(s) = g(s)for alls > c, thenf(t) =g(t)at alltwhere both are iff(t) andg(t) are piecewise continuous (continuous exceptat finite points where left and right limits exists,)]

CHAPTER 1 Laplace Transform Methods Laplace transform is a method frequently employed by engineers. By applying the Laplace transform, one can change an ordinary dif-

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