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