## Elementary Differential Equations and Boundary Value Problems 9th Edition

${Y(t)=t(A+Bt+Ct^2)+(C+Dt)cos(t)+(F+Et)sin(t)}$
Let $\;\;\;\;\;y=e^{rt}\\\\$ $y^{(4)}-{y}'''-{y}''-y=0 \;\;\;\;\Rightarrow \;\;\;\; r^4e^{rt}-r^3e^{rt}-r^2e^{rt}-e^{rt}=0\\\\$ $r^4-r^3-r^2-1=r(r-1)^2(r+1)=0$ $\rightarrow\;\;\;\;\; r_{1}=0\;\;\;\;\;\;\;or\;\;\;\;,r_{2,3}=1\;\;\;\;\;or\;\;\;\;r_{4}=-1\\\\$ $\boxed{y_{c}(t)= C_{1}+C_{2}e^{t}+C_{3}te^{t}+C_{4}e^{-t}}$ $\;\;\;\;g=(A+Bt+Ct^2)+(C+Dt)cos(t)+(F+Et)sin(t)$ $g_{1}=(A+Bt+Ct^2)\;\;\;\;\;$multiply this equation by $t$ $g_{1}=t(A+Bt+Ct^2)$ $g_{2}=(C+Dt)cos(t)+(F+Et)sin(t)$ $Y(t)=g_{1}+g_{2}$ $\boxed{Y(t)=t(A+Bt+Ct^2)+(C+Dt)cos(t)+(F+Et)sin(t)}$