Answer
$y(x)=C_1 e^{-3x}+C_2 \cos x+C_3 x \cos x +C_4x^2 \cos x+C_5 \sin x+C_6 x \sin x +C_7 x^2 \sin x $
Work Step by Step
Solve the auxiliary equation for the differential equation. $$r^7+3r^6+3r^5+9r^4+3r^3+9r^2+r+3=0$$
Factor and solve for the roots. $$(r+3)(r^2+1)^3=0$$
Roots are: $r_1=-3, r_2=-i; r_3=i$
This implies that there are $Seven$ independent solutions to the differential equation and the general equation is equal to $y(x)=C_1 e^{-3x}+C_2 \cos x+C_3 x \cos x +C_4x^2 \cos x+C_5 \sin x+C_6 x \sin x +C_7 x^2 \sin x $
