Problem $7$ Consider the following compartmental model for differentiation.

Where I is the input, $x_i$ is the concentration of the compartment, $f$ is the self$-$renewal fraction and $$ is the growth rate. The cell balance equation for this compartment is:

$\frac{d{x}_i}{dt}=I-x_i+2f{x}_i$

$=I-(1-2f)x_i$

$($A$)$ Evaluate $x_i(t)$ if $x_{\text{I}}(0)=x_0.$

$($B$)$ Does the time constant change with the self$-$renewal fraction $f?$

$($C$)$ Calculate the cell production rate as a function of time.

$($D$)$ Discuss why it is likely that $f_i>0$ for progenitor cell compartments.

$($E$)$ What happens if $f>\frac{1}{2}?$

Solve it by hand on paper please, I dont want an Ai answer.