= Solution
Expand the exact solution about $t_{n+1}=t$. The unscaled <local truncation error> is
$$
\begin{aligned}
d_h&=y(t+h)-y(t-h)-h\{a y'(t+h)+2(1-a)y'(t)+a y'(t-h)\}\\
&=\left(\frac13-a\right)h^3y'''(t)+\left(\frac1{60}-\frac a{12}\right)h^5y^{(5)}(t)+O(h^7).
\end{aligned}
$$
Only odd powers occur in this centered expansion. Unless $a=1/3$, the first nonzero coefficient is the $h^3$ coefficient, giving <order of a numerical method> two. For $a=1/3$, that term vanishes but the next coefficient is $-1/90$, giving <order of a numerical method> four. Thus
$$
\boxed{p=2\quad(a\ne1/3),\qquad p=4\quad(a=1/3).}
$$
These are exact orders for general smooth <ordinary differential equations>, since the respective first surviving derivatives need not vanish. With starting errors $O(h^p)$, the <zero-stability> established in part (a) makes the <global error> $O(h^p)$.
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