Answer
$y(x)=C_1e^{-x}+C_2e^{x}+C_3 e^x \cos x+C_4 x e^x \cos x+C_5 e^x \sin x+C_6 x e^x \sin x$
Work Step by Step
Solve the auxiliary equation for the differential equation. $$(r^2-2r+2)^2(r^2-1)=0$$
Factor and solve for the roots. $$(r^2-2r+2)^2(r-1)(r+1)=0$$
Roots are: $r_1=-1$, as a multiplicity of $1$ and $r_2=1$ as a multiplicity of $1$ and $r_3=1-i, r_4=1+i$ as a multiplicity of $2$.
This implies that there are five independent solutions to the differential equation.
Therefore, the general equation is equal to $y(x)=C_1e^{-x}+C_2e^{x}+C_3 e^x \cos x+C_4 x e^x \cos x+C_5 e^x \sin x+C_6 x e^x \sin x$
