Question

Find the solution of

(D^3+2D^2+D+2)y=0

Answer 1

$$(D^3+2D^2+D+2)y=0$$

The characteristic equation

$$k^3+2k^2+k+2=0$$

Roots

$$k_1=-2,\quad k_2=-i,\quad k_3=i$$

Solution of DE

$$y=C_1e^{-2x}+C_2e^{-ix}+C_3e^{ix}$$