Recurrence $u_{n+1}u_{n-1}-u_n^2=Ar^{n-1}$

I was going through some old puzzles and encountered one where there was a non-linear recurrence of form $$u_{n+1}u_{n-1}-u_n^2=Ar^{n-1}$$ with $u_0$ and $u_1$ given.

I know I can use a test form $u_n=p\alpha^n+q\beta^n$ with $\alpha \beta = r$, for example - so I can solve the recurrence.

My question is how to motivate such a step by manipulating the original recurrence - so that it becomes obvious, for example, that $u_n$ satisfies a linear recurrence? Is there a method other than guessing a solution?