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# set19 - \$ Set 19 Laplace Transform and IVPs Part 4 Kyle A...

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a39 a38 a36 a37 Set 19: Laplace Transform and IVPs Part 4 Kyle A. Gallivan Department of Mathematics Florida State University Ordinary Differential Equations Fall 2009 1

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a39 a38 a36 a37 Equivalence Example of time domain and Laplace domain equivalent operations Time Domain Laplace Domain linear combination linear combination differentiation multiplication by s multiplication by t differentiation translation by c multiplication by e cs multiplication by e ct translation by c What about products of Laplace transforms? 2
a39 a38 a36 a37 Convolution Integral Theorem 19.1. Let f ( t ) and g ( t ) be two functions whose Laplace transforms are F ( s ) and G ( s ) respectively for s>a . If H ( s ) = F ( s ) G ( s ) , s>a then h ( t ) = L 1 { H ( s ) } = integraldisplay t 0 f ( t τ ) g ( τ ) = integraldisplay t 0 f ( τ ) g ( t τ ) = f g 3

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a39 a38 a36 a37 General Form ay ′′ + by + cy = g ( t ) ,y (0) = y 0 , y (0) = y 0 arrowdblbothv Y ( s ) = ( as + b ) y 0 + ay 0 as 2 + bs + c + G ( s ) as 2 + bs + c = Φ( s ) + Ψ( s ) = Φ( s ) + H ( s ) G ( s ) where G ( s ) = L{ g } and H ( s ) = ( as 2 + bs + c ) 1 .
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set19 - \$ Set 19 Laplace Transform and IVPs Part 4 Kyle A...

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