Math Help - can any1 explain the roots of unity to me?

1. can any1 explain the roots of unity to me?

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.

2. The n roots of unity

Suppose you want the n roots of -1

e(int)= -1

t= pi/n cos( pi/n)+isin(pi/n) is one root to get the others divide 2pi/n and succesively add multiples of 2pi/n to pi/n to get the n roots

eg the 4 roots of -1 you use pi/4 ,3pi/4 etc in cos(t)+isin(t)

To get the n roots of 1

we have of course 0, then add 2pi/n to get the others

eg the four roots of unity are 0,pi/2,pi,3pi/2, correspondingly 1,i,-1,-1

3. Originally Posted by p00ndawg
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.
The characteristic equation, then, is $r^8+ r^4+ 16= 0$. Let $u= y^4$ and that becomes $u^2+ u+ 16= 0$ which you can solve with the quadratic formula. After you have u, you need to solve $y^4= u$. That's where the "fourth roots of unity" are concerned.