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