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Chapter 7Laplace TransformThe Laplace Transform can be used to solve di erential equations . Be-sides being a di erent and e cient alternative to variation of parame-ters and undetermined coe cients, theLaplace methodis particularlyadvantageous for input terms that are piecewise-de ned, periodic or Laplace transformor theLaplace integralof a functionf(t) de ned for 0 t <1is the ordinary calculus integration problemZ10f(t)e stdt;succinctly denotedL(f(t)) in science and engineering literature. TheL{notation recognizes that integration always proceeds overt= 0 tot=1and that the integral involves anintegratore stdtinstead of theusualdt. These minor di erences distinguishLaplace integralsfromthe ordinary integrals found on the inside covers of calculus Introduction to the Laplace MethodThe foundation of Laplace theory isLerch's cancellation lawR10y(t)e stdt=R10f(t)e stdtimpliesy(t) =f(t);orL(y(t) =L(f(t))impliesy(t) =f(t):(1)In di erential equation applications,y(t) is the sought-after unknownwhilef(t) is an explicit expression taken from integral , we illustrate Laplace 's method by solving the initial value prob-lemy0= 1; y(0) = 0:The method ob)}

Laplace Transform The Laplace transform can be used to solve di erential equations. Be-sides being a di erent and e cient alternative to variation of parame-ters and undetermined coe cients, the Laplace method is particularly advantageous for input terms that are piecewise-de ned, periodic or im-

  Equations, Transform, Laplace transforms, Laplace