Consider a cell composed of a cell body to which is attached a long, thin, tubular "process" $($e$.$g$.,$ the axon of a neuron or the flagellum of a sperm cell$)$ of length l cm $($Figure on right$).$ Suppose some substance n $($e$.$g$.,$ a metabolite$)$ is generated in the cell body and diffuses along the process, all along which the substance is consumed at a constant uniform rate, $\backslash$alpha n mol$/$s per unit length. The continuity equation for the substance thus becomes $$cn $$t $=\backslash$Phi n $$x $\backslash$alpha n A $,$ where A is the $($assumed constant$)$ cross$-$sectional area of the process. a$.$ Combine the modified continuity equation given above with Fick$$s law to obtain a modified form of the diffusion equation that must be satisfied by cn$.$ b$.$ Show that a solution of this equation in the steady state $($cn $$t $=\backslash$Phi n $$t $=0)$ is cn x $=\backslash$alpha n $2$DA x $2+$ a$0$x $+$ b$0$ and find values of the constants a$0$ and b$0$ corresponding to the boundary conditions cn o $=$ C$0$ and $\backslash$Phi n l $=09\_1$ Consider a cell composed of a cell body to which is attached a long, thin, tubular "process" $($e$.$g$.,$ the axon of a neuron or the flagellum of a sperm cell$)$ of length lcm $($Figure on right$).$ Suppose some substance n $($e$.$g$.,$ a metabolite$)$ is generated in the cell body and diffuses along the process, all along which the substance is consumed at a constant uniform rate, $\backslash$alpha $\_($n$)$mo$($l$)/($s$)$ per unit length. The continuity equation for the substance thus becomes $($delc$\_($n$))/($delt$)=-($del$\backslash$Phi $\_($n$))/($delx$)-(\backslash$alpha $\_($n$))/($A$),$ where A is the $($assumed constant$)$ cross$-$sectional area of the process. a$.$ Combine the modified continuity equation given above with Fick's law to obtain a modified form of the diffusion equation that must be satisfied by c$\_($n$).$ process b$.$ Show that a solution of this equation in the steady state $(($delc$\_($n$))/($delt$)=($del$\backslash$Phi $\_($n$))/($delt$)=0)$ is c$\_($n$)($x$)=(\backslash$alpha $\_($n$))/(2$DA$)$x$^(2)+$a$\_(0)$x$+$b$\_(0)$ and find values of the constants a$\_(0)$ and b$\_(0)$ corresponding to the boundary conditions c$\_($n$)($o$)=$C$\_(0)$ and $\backslash$Phi $\_($n$)($l$)=0.$