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The Inverse Laplace Transform

26 The Inverse Laplace TransformWe now know how to find Laplace transforms of unknown functions satisfying various initial-value problems. Of course, it s not the transforms of those unknown function which are usuallyof interest. It s the functions, themselves, that are of interest. So let us turn to the general issueof finding a functiony(t)when all we know is its Laplace transformY(s). Basic NotionsOn Recovering a Function from Its TransformIn attempting to solve the differential equation in , we gotY(s)=4s 3,which, sinceY(s)=L[y(t)]|sand4s 3=L[4e3t] s,we rewrote asL[y(t)]=L[4e3t].From this, seemed reasonable to conclude thaty(t)= , what if there were another functionf(t)with the same Transform as 4e3t?

The partial fraction expansion of this is Y(s) = 3s2 −28 (s − 4) s2 +4 = A s − + Bs +C s2 +4 for some constants A, B and C . There are many ways to find these constants. The basic method is to “undo” the partial fraction expansion by getting a common denominator and adding up the fractions on the right: 3s2 −28 ( s−4) s2 +4 = A ...

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