Answer
$y(x)=C_1 e^{-4x}+C_2 e^{4x}+C_3 e^{-2x} \cos 3x +C_4e^{-2x} \sin 3x $
Work Step by Step
Solve the auxiliary equation for the differential equation. $$r^4+4r^3-3r^2-64r-208=0$$
Factor and solve for the roots. $$(r-4)(r+4)(r^2+4r+13)=0$$
Roots are: $r_1=-4, r_2=4; r_3=-2-3i; r_4=-2+3i$
This implies that there are $\bf{3}$ independent solutions to the differential equation and the general equation is equal to $y(x)=C_1 e^{-4x}+C_2 e^{4x}+C_3 e^{-2x} \cos 3x +C_4e^{-2x} \sin 3x $
