Answer
${Y(t)=(A+Bt)e^{t}+Ctcos(2t)+Dtsin(2t)+Et^2}$
Work Step by Step
Let $\;\;\;\;\;y=e^{rt}\\\\$
${y}''''+4{y}''=0 \;\;\;\;\Rightarrow \;\;\;\; r^4e^{rt}+4r^2e^{rt}=0\\\\$
$r^4+4r^2=r(r^2-1)=r^2(r^2+4)=0 $
$ \rightarrow\;\;\;\;\; r_{1,2}=0\;\;\;\;\;\;\;or\;\;\;\;,r_{3,4}=\pm 2i\;\;\;\;\;\\\\$
$\boxed{y_{c}(t)= C_{1}+C_{2}t+C_{3}cos(2t)+C_{4}sin(2t)}$
$\;\;\;\;g=(A+Bt)e^{t}+Ccos(2t)+Dsin(2t)+E$
$g_{1}=Ccos(2t)+Dsin(2t)\;\;\;\;\;$multiply this equation by $t$
$g_{1}=Ctcos(2t)+Dtsin(2t)$
$g_{2}=(A+Bt)e^{t}$
$g_{3}=E\;\;\;\;\;$multiply this equation by $t^2$
$g_{3}=Et^2$
$Y(t)=g_{1}+g_{2}+g_{3}$
$\boxed{Y(t)=(A+Bt)e^{t}+Ctcos(2t)+Dtsin(2t)+Et^2}$
