LaplaceProperties

# LaplaceProperties - Laplace Transform Properties By...

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Laplace Transform Properties By building up some basic properties of the Laplace transform, we can expand the list of functions we know the transform of, thus increasing the number of IVP’s we can solve by this method. Property 1: If F ( s ) = L [ f ( x )] , then L [ - xf ( x )] = F 0 ( s ) . In words, multiplying by - x in our usual function space is the same as diﬀerentiation in transform space. I Example Find a function whose Laplace transform is 1 ( s - a ) 2 . Solution We have that d ds ± - 1 s - a ² = 1 ( s - a ) 2 Furthermore, we know that L [ - e ax ] = - 1 s - a so by Property 1 we have L [ xe ax ] = L [ - x ( - e ax )] = d ds ± - 1 s - a ² = 1 ( s - a ) 2 .

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Property 2: If F ( s ) = L [ f ( x )] , then L [ e ax f ( x )] = F ( s - a ) . In words, multiplying by e ax in our usual function space is the same as translation to the right by a in transform space. I Example Find a function whose Laplace transform is 1 s 2 - 4 s + 9 . Solution
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## This note was uploaded on 03/03/2012 for the course MATH 4430 at Colorado.

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LaplaceProperties - Laplace Transform Properties By...

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