L73 - Fall 2003 Math 308/501502 7 Laplace Transforms 7.3...

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Fall 2003 Math 308/501–502 7 Laplace Transforms 7.3 Properties of Laplace Transforms Fri, 10/Oct c ± 2003, Art Belmonte Summary Looking up Laplace transforms in tables is aided by employing various properties that the Laplace transform satisfies. This also aids a computer when calculating transforms! Linearity Property For constants α and β and piecewise continuous functions f and g of exponential order, we have L { α f + β g } = α L { f } + β L { g } . Laplace transforms of derivatives Let y ( k ) , k = 0 , 1 ,... n - 1, be piecewise differentiable and continuous functions of exponential order. Let y ( n ) be piecewise continuous and of exponential order. If L { y ( t ) } ( s ) = Y ( s ) ,then L n y ( n ) o ( s ) = s n Y ( s ) - n X k = 1 s n - k y ( k - 1 ) ( 0 ) In particular, for n = 1, we have L ± y 0 ² ( s ) = sY ( s ) - y ( 0 ), and for n = 2, this yields L ± y 00 ² ( s ) = s 2 Y ( s ) - sy ( 0 ) - y 0 ( 0 ). First translation or shifting property Let f be a piecewise continuous function of exponential order with transform L { f ( t ) } ( s ) = F ( s ) and let c be a constant. Then L ± e ct f ( t ) ² ( s ) = F ( s - c ) for s > c Second translation or shifting property If L { f } ( s ) = F ( s ) and g ( t ) = ³ f ( t - a ), t > a 0 , t < a L { g ( t ) } ( s ) = G ( s ) = e - as F ( s ) Change of scale property If L { f } ( s ) = F ( s ) L { f ( at ) } ( s ) = 1 a F ´ s a µ Relation between Laplace transform and its derivatives Let f be a piecewise continuous function of exponential order with transform L { f ( t ) } ( s ) = F ( s ) .Then L ± t n f ( t ) ² ( s ) = ( - 1 ) n F ( n ) ( s ) In particular, for n = 1wehave L { tf ( t ) } ( s ) =- F 0 ( s ) Hand Examples Example A Compute the Laplace transform of t n for nonnegative integers n .
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This note was uploaded on 03/29/2008 for the course MATH 308 taught by Professor Comech during the Spring '08 term at Texas A&M.

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L73 - Fall 2003 Math 308/501502 7 Laplace Transforms 7.3...

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