Consider the IVP $$ y'(t)=f(t,y(t)), \quad \quad y(0)=y_0. $$ How can I show that the following multistep method has consistency order 3: $$ y_{k+2}+4y_{k+1}-5y_k=h(4f(t_{k+1},y_{k+1})+2f(t_k,y_k))? $$ Also: is the method zero-stable?
How would I go about this question? I know the definition of the truncation error and consistency order, but don't I need the exact solution to calculate both?
Yes, you insert the Taylor expansions of an exact solution and look which terms cancel and what the lowest degree is where the terms do not cancel.
That does not mean that you need to procure the actual exact solution, you work with general, theoretical, abstract objects. The $f$ is general in the property that unique solutions of sufficient smoothness exist, and $y$ is such a solution that has Taylor polynomials of a high enough degree to establish the order of the method.