# How to solve this non-linear second order difference equation

Is there a direct method to analytically solve this non linear second order difference equation problem ?

$$$$\begin{split} \left \{ \begin{array}{ll} \frac{\beta\nu}{2\sqrt{\Phi_{t+1}}} - 2\alpha (\Phi_{t+1} - \Phi_{t}) + \beta 2\alpha (\Phi_{t+2} - \Phi_{t+1}) = 0 \\ \Phi_0 \ge 0 \\ \Phi_T \ge 0 \end{array} \right. \end{split}$$$$ with $$\beta \in ] 0, 1 ]$$, $$\nu >0$$ and $$\alpha >0$$ .

I am able to approximate the solution through a backward shooting method by rewriting the problem as : $$$$\begin{split} \left \{ \begin{array}{ll} \Phi_t = \Phi_{t+1} - \frac{\beta}{2 \alpha} \left [ \frac{\nu}{2\sqrt{\Phi_{t+1}}} + 2\alpha (\Phi_{t+2} - \Phi_{t+1}) \right ] \\ \Phi_0 \ge 0 \\ \Phi_T \ge 0 \end{array} \right. \end{split}$$$$ but I wonder if there is a way to have the analytical solution of this problem ?