Consider a multistep method for numerical solution of the differential equation $\mathbf{y}^{\prime}=\mathbf{f}(t, \mathbf{y})$ :

$$\mathbf{y}_{n+2}-\mathbf{y}_{n+1}=h\left[(1+\alpha) \mathbf{f}\left(t_{n+2}, \mathbf{y}_{n+2}\right)+\beta \mathbf{f}\left(t_{n+1}, \mathbf{y}_{n+1}\right)-(\alpha+\beta) \mathbf{f}\left(t_{n}, \mathbf{y}_{n}\right)\right],$$

where $n=0,1, \ldots$, and $\alpha$ and $\beta$ are constants.

(a) Define the order of a method for numerically solving an ODE.

(b) Show that in general an explicit method of the form $(*)$ has order 1 . Determine the values of $\alpha$ and $\beta$ for which this multistep method is of order 3 .

(c) Show that the multistep method (*) is convergent.