Implicit Linear multistep method order?

Question

Considering the following linear multistep method:

$y_{k+2} = y_{k+1} + \frac{h}{12} \left( -f(x_{k},y_{k}) + 8f(x_{k+1},y_{k+1})+ 5f(x_{k+2},y_{k+2}) \right)$

What is it's order? What is the maximum order that this method can attain?

Answer 1

To determine an upper bound for the order, insert $f(x,y)=y$ and $y_k=e^{kh}$ and compute the difference of both sides. $$ e^{2h}-e^h-\frac{h}{12}(-1+8e^h+5e^{2h}) $$ and then insert the Taylor series of the exponential.