The first$-$order irreversible reactions: are being carried out in an ideal, isothermal PFR$.$ At the reactor temperature, $k_1=k_2=1\mathrm{.}0s^{-1}$ and $k_3=2\mathrm{.}0s^{-1}.$ The feed to the reactor contains $C_{A0}=2\mathrm{.}0mo\frac{l}{d}{m}^3$ and no D$,$ S$,$ or U$.$

$($a$)(10$ pts$)$ What is the instantaneous selectivity $S_{\frac{D}{U}}?$ Is it a constant along the reactor bed?

$($b$)(30$ pts$)$ Develop a set of coupled ODEs in terms of molar flow rates: $F_A,F_D,F_S,F_U,$ and $F_T,$ for this case. Please include all the given parameters in your equations.