Chapter 9.3, Problem 43E

A glucose solution is administered intravenously into the bloodstream at a constant rate r. As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate that is proportional to the concentration at that time. Thus a model for the concentration C = C(t) of the glucose solution in the bloodstream

$\frac{dC}{dt}=r-kC$

where k is a positive constant.

(a) Suppose that the concentration at time t = 0 is C_o. Determine the concentration at any time t by solving the differential equation.

(b) Assuming that C₀ < r/k, find lim_t→∞ C(t) and interpret your answer.