Answer
$y(x)=C_1 e^{-4x}+C_2 \cos 5x+C_3 x \cos 5x +C_4 \sin 5x+C_5 x \sin 5x $
Work Step by Step
Solve the auxiliary equation for the differential equation. $$r^5+4r^4+50r^3+200r^2+625r+2500=0$$
Factor and solve for the roots. $$(r+4)(r^2+25)^2=0$$
Roots are: $r_1=-4, r_2=-5i; r_3=5i$
This implies that there are $Five$ independent solutions to the differential equation and the general equation is equal to $y(x)=C_1 e^{-4x}+C_2 \cos 5x+C_3 x \cos 5x +C_4 \sin 5x+C_5 x \sin 5x $
