I am interested in the following first-order rational difference equation (recurrence relation): $$x_{n+1} = x_n + \dfrac{\alpha}{x_n} + \beta \ ; \ x_0 >0$$ for positive $\alpha$ and $\beta$. Is there a general solution for this kind of difference equation? Or how can I approximate it?
It clearly is strictly increasing and a solution should be roughly linear for large n. Since it is a rational difference equation I tried different substitutions like $x_n = \dfrac{p_n}{q_n}$ but could not come up with a system of difference equations that appears to be solveable.