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 - 4.3 The Method of Undetermined Coefficients - Problems - Page 237: 14

Answer

$Y(t)=t(At+B)e^{-t}+Ccos(t)+Dsin(t)$

Work Step by Step

Let $\;\;\;\;\;y=e^{rt}\\\\$ ${y}'''-2{y}'=0 \;\;\;\;\Rightarrow \;\;\;\; r^3e^{rt}-re^{rt}=0\\\\$ $r^3-r=r(r^2-1)=r(r-1)(r+1)=0 $ $ \rightarrow\;\;\;\;\; r_{1}=0\;\;\;\;\;\;\;or\;\;\;\;,r_{2}=1\;\;\;\;\;or\;\;\;\;r_{3}=-1\\\\$ $\boxed{y_{c}(t)= C_{1}+C_{2}e^{t}+C_{3}e^{-t}}$ $\;\;\;\;g=(At+B)e^{-t}+Ccos(t)+Dsin(t)$ $g_{1}=(At+B)e^{-t}\;\;\;\;\;$multiply this equation by $t$ $g_{1}=t(At+B)e^{-t}$ $g_{2}=Fe^{t}\;\;\;\;\;$multiply this equation by $t^2$ $g_{2}=Ccos(t)+Dsin(t)$ $Y(t)=g_{1}+g_{2}$ $\boxed{Y(t)=t(At+B)e^{-t}+Ccos(t)+Dsin(t)}$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.