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Unformatted text preview: Laplace Transform March 24, 2011 This page aims to recap our minilecture on Laplace transform. The Laplace transform of a function f ( t ) , if it exists, is given by L { f ( t ) } = Z e st f ( t ) dt = F ( s ) . (1) Furthermore, the original function in Eq. ( 1 ) is called the inverse transform of F ( s ) , and denoted by f ( t ) = L 1 { F ( s ) } . (2) Keep in mind that a function of t becomes a function of s under laplace transform. We will utilize the Laplace transform to solve initial value problems using the steps summarized next. Problem statement: Find the solution the the IVP y 00 ( t ) + Ay ( t ) + By ( t ) = g ( t ) , given y (0) , y (0) . (3) Steps: 1. Take Laplace transform of both sides of the differential equation L { y 00 ( t ) } + L { Ay ( t ) } + L { By ( t ) } = L { g ( t ) } , and recall that L { y ( t ) } = Y ( s ) , (4) L { y ( t ) } = sY ( s ) y (0) , (5) L { y 00 ( t ) } = s 2 Y ( s ) sy (0) y (0) . (6) Therefore, the resulting expression is s 2 Y (...
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This note was uploaded on 05/02/2011 for the course GE 207K taught by Professor None during the Spring '10 term at University of Texas at Austin.
 Spring '10
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