Problem 2. Consider the following compartmental model of differentiation shown in the scheme below. This is an analytical problem requiring mathematical manipulations only, so show all math. Parameters: I is the input flux (constant), X is the cell concentration in compartment i, fis the fraction of self-renewal and u is the specific growth rate. (a) Derive the cell balance equation for compartment that describes the above process (b) Obtain an expression for X(t) if X{t=0) = Xo (c) What is the time constant, t, for this process? (d) Obtain an expression for the cell production rate, dx/dt as a function of time, t. Note this expression should include all constant parameters and initial cell concentration. (e) What happens if f>1/2? Iff=1/2