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Unformatted text preview: Chapter 2 The Laplace Transform Objective: Mathematical preparation for the study of next chapters ontents Contents: 2.1 Introduction 2 Complex Numbers and Harmonic Motion 2.2 Complex Numbers and Harmonic Motion 2.3 Laplace Transformation 2.4 Inverse Laplace Transformation 2.5 Solving Linear Differential Eqs. by Laplace Transform JC 2007 1 .3 Laplace Transform 2.3 Laplace Transform Why do we need Laplace Transform? InitialValue Problems ODE's or PDE's Laplace Transform Algebra Problems olutions of olutions of verse Laplace Transform VERY EASY ! DIFFICULT ! Solutions of ODE's or PDE's Solutions of Algebra Problems Inverse Laplace Transform JC 2007 2 olution by Undetermined Coefficients (1) Solution by Undetermined Coefficients (1) Example: 4 01 . = + x x Step 1: Solve homogeneous equation to get complimentary 70 ) ( = x 1 = solution Xc Homogeneous Equ. t c Ce x = 01 . + x x Assume o oge eous qu (harmonic motion ) Then t c t c x e C x 2 2 = = c c x e C x = = 1 = t e From substitution JC 2007 3 ) 01 . ( + Ce olution by Undetermined Coefficients (2) Solution by Undetermined Coefficients (2) ) 01 . ( = + t Ce From substitution 1 haracteristic Equ 01 . = + Characteristic Equ. 01 . = Characteristic root (eigenvalue) t c Ce x 01 . = Complementary solution: JC 2007 4 olution by Undetermined Coefficients (2) Solution by Undetermined Coefficients (2) Step 2: Solve nonhomogeneous equation to get particular solution Xp 4 01 . = + x x Assume NonHomogeneous Equ. D x p = (Xp takes the form of forcing function and all of its derivatives ) 4 01 . = D Substitution 400 = p x Particular solution: JC 2007 5 olution by Undetermined Coefficients (2) Solution by Undetermined Coefficients (2) Step 3: Total solution=Xc+Xp, Using IC to determine constants. t c Ce x 01 . = 400 = p x 70 ) ( = x JC 2007 6 Total solution: t e x 01 . 330 400 = ow Solution by Laplace Transform Works (1)? How Solution by Laplace Transform Works (1)?...
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This note was uploaded on 12/08/2008 for the course ME 3504 taught by Professor Tschang during the Fall '08 term at Virginia Tech.
 Fall '08
 TSChang
 Laplace

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