Let $(u_n)_{n\geq 0}$ a sequence of reals satisfying the following recurrence relation:
$$ \forall n>1, \qquad u_{n+2} = r^{n+2}\big( Au_{n+1}+ Bu_n\big) $$
for fixed non zero constants $A,B,r$. Is there a closed form expression of the (2 dimensional) space of solutions, parametrised by the initial values $u_0, u_1$?
My observation is that the set of solution is a 2 dimensional vector space, so that we would only need to find the solutions for the initial parameters $(0,1)$ and $(1,0)$ and then express the generic solution as a linear combination of these twos. Point is is can't find a sympathetic expression for the elements of this bases.
Look for solutions of the form $u_n=a^n$ for some (in general complex) number $a\ne0$. You will get a quadratic equation for $a$. If it has two different solutions $a_1$ and $a_2$, then $a_1^n$ and $a_2^n$ are two linearly independent solutions and the general solution is $u_n=C_1\,a_1^n+C_2\,a_2^n$. I leave to you the case when there is only one solution to the quadratic equation.