i have my book obviously but its extremely unclear to me as to how to use the method to acquire roots other than guessing.

for example

y(8) + 8y(4) + 16y = 0

how do i get the roots for my homogenous using roots of unity?

the y(x)<- whatever inside parenthesis is to what prime it is. So for the first one it is y with the derivative taken 8x.