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 $t^2$
$g_{2}=Ccos(t)+Dsin(t)$
$Y(t)=g_{1}+g_{2}$
$\boxed{Y(t)=t(At+B)e^{-t}+Ccos(t)+Dsin(t)}$
