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Unformatted text preview: The Laplace Transform We begin by defining a certain linear operator called the Laplace transform, which we will use to turn the problem of solving a second order differential equation into the much simpler problem of solving an algebraic equation. Definition The Laplace transform is the linear operator L defined by L [ f ( x )] = F ( s ) = Z e- sx f ( x ) dx Remarks (1) Note that if we are given a function f ( x ) that we wish to transform, F ( s ) will only be defined at those points s where the corresponding indefinite integral converges. (2) We will only consider functions f ( x ) that are defined on x &lt; so that the integral in the Laplace transform makes sense. I Example Find the Laplace transform of the function f ( x ) = e ax . Solution From the definition of the Laplace transform we compute L [ e ax ] = Z e- sx e ax dx = Z e ( a- s ) x dx = lim b Z b e ( a- s ) x dx = lim b e ( a- s ) x a- s b = lim b e ( a- s ) b a- s- 1 a- s = ( 1 s- a...
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