$7\mathrm{.}3.($a$)$ Show experimentally or algebraically that the BS$23$ algorithm is exact for f$($t$,$y$)=1,$ f$($t$,$y$)=$ t$,$ and f$($t$,$y$)=$ t$2,$ but not for f$($t$,$y$)=$ t$3.$

$($b$)$ When is the ode$23$ error estimator exact?