General solution to 3rd order recurrence relation

Hi, I'm a bit stuck on this question:

For a fixed *r*, show that the general solution to the recurrence relation

$\displaystyle x_{n} = rx_{n-1} + r^2x_{n-2} - r^3x_{n-3} $

is given by

$\displaystyle x_{n} = Ar^n + Bnr^n + C(-r)^n $ for constants $\displaystyle A, B, C \in \mathbb{R}$

I've done this so far:

Let $\displaystyle x_{n-1} = a^n$

$\displaystyle a^n = ra^{n-1} +r^2na^{n-2} -r^3xa^{n-3} $

Divide through by $\displaystyle a^{n-3}$

$\displaystyle a^3 = ra^2 + r^2na - r^3$

But I'm stuck now on getting those powers back to $\displaystyle n$. Any pointers most welcome!