# Math Help - Diff EQ Medication Equilibrium Word Problem

1. ## 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 $k=-0.347$.

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

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

(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: $Q=2.5e^{-.347t}$

**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.

2. Originally Posted by Jenberl
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 $k=-0.347$.

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

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

(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: $Q=2.5e^{-.347t}$

**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.
Your answer to (b) is wrong. You have not written down the correct differential equation. You forgot to consider the fact that morphine is also entering the body.

You have to solve $\frac{dQ}{dt} = {\color{red}2.5} - 0.347 Q$

subject to $Q(0) = 0$.

If the amount of morphine in the body reaches equilibrium then $\frac{dQ}{dt} \rightarrow 0$ and so $2.5 - 0.347 Q \rightarrow 0 \Rightarrow Q \rightarrow \frac{2.5}{0.347}$.

Note: This is the limiting (ie. equilibrium) value that your solution for Q must approach as $t \rightarrow + \infty$ and so you can use it as a check of your final answer.