I am trying to solve a difference equation, but once I get a certain formula, I do not know how to reform it. There is a solution, but it does not explain how to reform the equation.
The image of the problem and the solution
I get to this point $ (\alpha-\beta)*Y_{t+1} = -\beta Y_{t} $, but how do I proceed in order to get the formula from the result?
This is just a matter of algebraic manipulation:
$$-\frac\beta{\alpha-\beta}=\frac\beta{\beta-\alpha}=\frac{\beta-\alpha+\alpha}{\beta-\alpha}=\frac{\beta-\alpha}{\beta-\alpha}+\frac\alpha{\beta-\alpha}=1+\frac\alpha{\beta-\alpha}$$