I have an exercise that I do not understand. We have to solve an initial value problem:
$$ \begin{array}{ccl} y'(t) &=& f(t,y(t)) \\ y(a) &=& y_0 \end{array} $$
We have to derive an effective method to calculate $\alpha$, $\beta$ and $\gamma$ with the highest possible accuracy. The method has the form below: $$ y_{k+1} = \alpha y_k + \beta y_{k-1} + \gamma h f(t_k, y_k) $$ We can even assume that $y_1 = y(a+h)$ and that the interval is equidistant.
I can guess that we have to derive a Euler forward, but I cannot solve it? Besides, they ask the order of accuracy of the found method. Shall I write all the $y(a+h)$ as a Taylor polynomial? I would greatly appreciate some guidance.