(a) For the equation $y^{\prime}=f(t, y)$, consider the following multistep method with $s$ steps,

$$\sum_{i=0}^{s} \rho_{i} y_{n+i}=h \sum_{i=0}^{s} \sigma_{i} f\left(t_{n+i}, y_{n+i}\right)$$

where $h$ is the step size and $\rho_{i}, \sigma_{i}$ are specified constants with $\rho_{s}=1$. Prove that this method is of order $p$ if and only if

$$\sum_{i=0}^{s} \rho_{i} Q\left(t_{n+i}\right)=h \sum_{i=0}^{s} \sigma_{i} Q^{\prime}\left(t_{n+i}\right)$$

for any polynomial $Q$ of degree $\leqslant p$. Deduce that there is no $s$-step method of order $2 s+1$.

[You may use the fact that, for any $a_{i}, b_{i}$, the Hermite interpolation problem

$$Q\left(x_{i}\right)=a_{i}, \quad Q^{\prime}\left(x_{i}\right)=b_{i}, \quad i=0, \ldots, s$$

is uniquely solvable in the space of polynomials of degree $2 s+1 .]$

(b) State the Dahlquist equivalence theorem regarding the convergence of a multistep method. Determine all the values of the real parameter $a \neq 0$ for which the multistep method

$$y_{n+3}+(2 a-3)\left[y_{n+2}-y_{n+1}\right]-y_{n}=h a\left[f_{n+2}+f_{n+1}\right]$$

is convergent, and determine the order of convergence.