Answer
$Y(t)=At^2e^{t}+Bcos(t)+Csin(t)$
Work Step by Step
Let $\;\;\;\;\;y=e^{rt}\\\\$
${y}''''-2{y}''+y=0 \;\;\;\;\Rightarrow \;\;\;\; r^4e^{rt}-2re^{rt}+e^{rt}=0\\\\$
$r^4-2r^2+1=r(r^2-1)=(r-1)^2(r+1)^2=0 $
$ \rightarrow\;\;\;\;\; r_{1,2}=1\;\;\;\;\;\;\;or\;\;\;\;,r_{3,4}=-1\;\;\;\;\;\\\\$
$\boxed{y_{c}(t)= C_{1}e^{t}+C_{2}te^{t}+C_{3}e^{-t}+C_{4}te^{-t}}$
$\;\;\;\;g=Ae^{t}+Bcos(t)+Csin(t)$
$g_{1}=Ae^{t}\;\;\;\;\;$multiply this equation by $t^2$
$g_{1}=At^2e^{t}$
$g_{2}=Bcos(t)+Csin(t)\;\;\;\;\;$
$Y(t)=g_{1}+g_{2}$
$\boxed{Y(t)=At^2e^{t}+Bcos(t)+Csin(t)}$
