wksht06sol

# wksht06sol - MATH 152 L1& L2 Applied Linear Algebra and...

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Unformatted text preview: MATH 152 L1 & L2 Applied Linear Algebra and Differential Equations Spring 2004-05 ¥ Week 06 Worksheet Solutions : Laplace Transform Q1. (Initial value problem) Consider the initial value problem y- y = e 2 t , y (0) = 1. (a) Show that L{ y ( t ) } = s L{ y ( t ) } - y (0). Solution The formula follows from integration by parts, L{ y ( t ) } def. = Z ∞ e- st y ( t ) dt = Z ∞ e- st d ( y ( t )) = e- st y ( t ) fl fl fl t = ∞ t =0 + s Z ∞ e- st y ( t ) dt =- y (0) + s L{ y ( t ) } . (b) Apply the Laplace transform to both sides of the given equation, and then determine L{ y ( t ) } . Solution Applying the Laplace transform to the differential equation, we have L{ y- y } = L{ e 2 t } . By the linearity of L , we have L{ y } - L{ y } = L{ e 2 t } , [ s L{ y } - y (0)]- L{ y } = 1 s- 2 , ( s- 1) L{ y } = 1 s- 2 + 1 = s- 1 s- 2 , L{ y ( t ) } = 1 s- 2 . (c) Apply the inverse Laplace transform to find y ( t ) from L{ y ( t ) } ....
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## This note was uploaded on 09/30/2010 for the course MATH MATH152 taught by Professor Kcc during the Spring '10 term at HKUST.

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wksht06sol - MATH 152 L1& L2 Applied Linear Algebra and...

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