Answer
$y(x)=C_1e^{-x}\cos 3x+C_2 e^{-x}\sin 3x+C_3xe^{-x}\cos 3x+C_4xe^{-x} \sin 3x$
Work Step by Step
Solve the auxiliary equation for the differential equation. $$(r^2+2r+10)^2=0$$
Factor and solve for the roots.
Roots are: $$r_1=-1-3i, r_2=-1+3i$$
This implies that there are two independent solutions to the differential equation $y_1(x)=e^{-x}\cos 3x$ and $y_2= e^{-x}\sin 3x$ and $y_3=xe^{-x}\cos 3x$ and $y_4=xe^{-x} \sin 3x$
Therefore, the general equation is equal to $y(x)=C_1e^{-x}\cos 3x+C_2 e^{-x}\sin 3x+C_3xe^{-x}\cos 3x+C_4xe^{-x} \sin 3x$
