THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

Appendix CUsing Laplace Transforms to SolveDifferential EquationsC.1 IntroductionNot only all aspects of the of Laplace transforms will be presented, but also enoughof the characteristics will be presented, discussed here to help the reader understandand appreciate the elegance and powerfulness of the procedures that canbe effectively applied to solve linear differential equations. The essential idea ofLaplace transforms is that they are used to convert a differential equation into analgebraic one in order to solve the differential equation.Let f (t) be a function of t for t > 0. Its Laplace transform is defined asF(s) =∫ ∞0f (t)e −st dt = L[ f (t), s]where s is a complex variable which can be expressed as(C.1)s = α + iβ(C.2)The function f (t) is Laplace transformable for α>0if∫ TlimT →∞ 0| f (t)|e −αt dt < ∞If f (t) = A (a constant), for t > 0, the Laplace transform of the constant A isL[A] =∫ ∞0Ae −st dt =− 1 s e−st∣ ∣ ∞ 0We therefore have the Laplace transform pair= A s = F(s)f (t) = A ⇔ F(s) = A sIf f (t) = e −at for t > 0, then the Laplace transform of this exponential functionisLe −at =∫ ∞0e −at e −st dt =∫ ∞0e −(s+a)t =As + a637