Elementary Differential Equations and Boundary Value Problems 9th Edition

Published by Wiley
ISBN 10: 0-47038-334-8
ISBN 13: 978-0-47038-334-6

Chapter 4 - Higher Order Linear Equations - 4.2 Homogenous Equations with Constant Coefficients - Problems - Page 232: 17

Answer

$y=[C_{1}e^{t}+C_{2} te^{t}+C_{3}t^2e^{t}] + [C_{4}e^{-t}+C_{5}te^{- t} + C_{6}t^2e^{-t}]$

Work Step by Step

Let $\;\;\;\;\;y=e^{rt}\\\\$ $y^{(6)}-3y^{(4)}+3{y}''-y=0 \;\;\;\;\;\Rightarrow \;\;\;\; r^6e^{rt}-3r^4e^{rt}+3r^2e^{rt}-e^{rt}=0\\\\$ $r^6-3r^4+3r^2-1=(r-1)^3(r+1)^3=0 \;\;\;\;$$\rightarrow \;\;\;\; r_{1},r_{2},r_{3}=1\;\;\;\;or\;\;\;\; r_{4}=-1,r_{5}=-1,r_{6}=-1\\\\$ $y=[C_{1}e^{t}+C_{2} te^{t}+C_{3}t^2e^{t}] + [C_{4}e^{-t}+C_{5}te^{- t} + C_{6}t^2e^{-t}]$
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