Find the solution of the difference equation $y_{k + 2} - 5y_{k + 1} + 6y_{k} = 2$ with $y_{0} = 1$ and $y_{1} = 2$ using generating functions.
My attempt:
First of all, we need find $$Y(s) = \sum_{k \geq 0}y_{k}s^{k}$$
Thus, we have that
$$\sum_{k \geq 0}y_{k+2}s^{k} - 5\sum_{k\geq 0}y_{k + 1}s^{k} + 6\sum_{k\geq 0 }y_{k}s^{k} = 2\sum_{k \geq 0}s^{k}$$
$$\Rightarrow \frac{Y(s) - 1 - 2s}{s^{2}} - 5\left(\frac{Y(s) - 1}{s}\right) + 6Y(s) = \frac{2}{1-s}$$
Now we can isolate $Y(s)$. However, I would need to expand $Y(s)$ in power series, but I'm having trouble. Someone can help me? Thank you in advance.
After solving for $Y(s)$, use partial fraction decomposition to rewrite as $$Y(s)=\frac{1}{1 - s} - \frac{1}{1 - 2 s} + \frac{1}{1 - 3 s}.$$ Now recognize each term as a geometric series: $$Y(s)=\sum_{k\ge 0} s^k - \sum_{k\ge 0}(2 s)^k + \sum_{k\ge 0}(3 s)^k,$$ which implies that $$y_k=1 - 2^k + 3^k.$$