Consider the following multistep method $$y_n-y_{n-3} + \alpha y_{n-1}-\alpha y_{n-2}=h(\beta f_{n-1}+\beta f_{n-2})$$ where $\alpha$ and $\beta$ are undetermined constants.
How would one go about choosing constants $\alpha$ and $\beta$ s.t. the truncation error of the method is $O(h^4)$? Moreover, how would you do this in general?
I know that given a general linear $m$-step method, $$\sum_{n=0}^ma_ny_{k+n}=h\sum_{n=0}^mb_nf(t_{k+n},y_{k+n}),$$ the local truncation error is of order $p\geq 1$ iff $$\sum_{n=0}^ma_n=0\quad\text{and}\quad \sum_{n=0}^m n^{k}a_n=k\sum_{n=0}^mn^{k-1}b_n$$ for $k=1,\ldots,p$.
However, I'm not sure how to recover the constants from the step where $$\sum_{n=0}^m n^{k}a_n\ne k\sum_{n=0}^mn^{k-1}b_n.$$