I'm trying to find the auxiliary roots of this DE
So the auxiliary eqn is
and m=i ?
Use De Moivre's formula to find the three roots.
EDIT:
for k = 0, 1, and 2
When k=0,
when k =1,
and for the k=2,
The complex roots are not conjugate pairs like in the other problem, so the general solution is
BTW, Wolfram Alpha (the web site associated with Mathematica) can solve differential equations like these.
It should be
Then
But the rest is correct.
I used De Moivre's formula to find the roots for the other problem, too.
so
Then let k =0,1,2,and 3 to find the four roots.
http://en.wikipedia.org/wiki/De_Moivre's_formula
It's because of the location of -i on the complex plane.
I'll try to explain, but you should really study up a bit on complex numbers.
All real numbers (a+0i) are located along the x axis, all purely imaginary numbers (0+bi) are located along the y axis, and all complex numbers (a+bi) are located everywhere else.
is located on the negative y axis and can be represented as a vector of unit length (since ) with it's tail at the origin and it's head at y=-1 .
The angle (going counterclockwise) that the vector makes with the positive horizontal axis is . But you usually want the angle to be between and . The angle in that range that is equal to is
You can write any number (real, imaginary, or complex) in the form where is the distance of the point from the origin in the complex plane and is the angle formed by the vector and the positive x axis.
So
For the other problem the angle is because -1 is located on the negative x axis.