When an individual has asthma, an inhaler can be used to widen the lower airways, which sprays an amount of pharmaceutical $($salbutamol$)$ down the airways. However, during transport in the trachea much of the salbutamol might be depositing on the walls and it is interesting for pharmaceutical manufacturers how much salbutamol is$"$ lost" in the trachea. Assume that the trachea is a straight, rigid tube with length $L(m)$ and radius $R(m)$ and during the small burst of the inhaler the velocity, $v_t(\frac{m}{s}),$ can be assumed constant. The salbutamol bolus with a changing concentration $c_{\text{s}\text{a}\text{l}}(\frac{\text{m}\text{o}\text{l}}{m^3})$ can be assumed cylindrical with radius $R$ and length $y(m).$ The deposition of salbutamol onto the trachea wall can be described by$,q=.c_{\text{s}\text{a}\text{l}},$ in $mol{s}^{-1}{m}^{-2}.$ which is the mass flux per unit area. Assume that the amount of salbutamol is perfectly mixed due to turbulence caused by the high speed.

Motion of an amount of salbutamol through the trachea

$($a$)$ Derive an ODE describing the spatial evolution $($in terms of axial coordinate y$)$ of the concentration of the salbutamol bolus, $c_{\text{s}\text{a}\text{l}\text{.}}.$