Diff EQ Medication Equilibrium Word Problem

Morphine is often used as a pain-relieving drug. The half-life of morphine

in the body is 2 hours. Suppose that morphine is administered to a patient intravenously at a rate of 2.5 mg per hour, and the rate at which the morphine is eliminated is proportional to the amount present.

(a) Show that the constant of proportionality for the rate at which morphine leaves the body (in mg/hr) is $\displaystyle k=-0.347$.

$\displaystyle .5Q_{0}=Q_{0}e^{2k}$

$\displaystyle k=\frac{ln(.5)}{2}$ then, $\displaystyle k=-.347$

(b) Write a differential equation for the quantity Q of morphine in the blood

after t hours. Solve the differential equation from (a).

So, I got: $\displaystyle Q=2.5e^{-.347t}$

**My query is about the following:

**c) If we wait long enough, the amount of morphine is the blood will approach an “equilibrium solution” where there is no change**

in the amount Q. Use the differential equation to find the equilibrium solution. (This is the long-term amount of morphine in the body, once the system has stabilized.)

I'm unsure of how to create the equilibrium solution, because I am not even sure that my answer for "(b)" is correct.