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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 is

$ \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 $ C_o < r/k, $ find lim $ _{t \to \infty} C(t) $ and interpret your answer.

a) $$C(t)=\left(C_{0}-r / k\right) e^{-k t}+r / k$$

b) $$\lim _{t \rightarrow \infty} C(t)=r / k$$

Differential Equations

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