Answer
$y=[C_{1}+C_{2} t] + [C_{3}e^{t}+C_{4}e^{- t}] + [C_{5}cos( t)+C_{6}sin( t)]$
Work Step by Step
Let $\;\;\;\;y=e^{rt}\\\\$
$y^{6}+{y}''=0 \;\;\; \Rightarrow \;\;\;\;\; r^6e^{rt}+r^2e^{rt}=0\\\\$
$r^2(r^4-1)=r^2(r^2-1)(r^2+1) \;\;\;\; \rightarrow \;\;\;\; r_{1},r_{2}=0\;\;\;\;or\;\;\;\; r_{3}=1\;,r_{4}=-1\;\;\;or\;\;\;\;r_{5}=i,r_{6}=-i\\\\$
The general solution for complex roots is:
$y= C_{1}e^{\alpha t}cos(\beta t)+C_{2}e^{\alpha t}sin(\beta t)$
$y=[C_{1}+C_{2} t] + [C_{3}e^{t}+C_{4}e^{- t}] + [C_{5}cos( t)+C_{6}sin( t)]$
