Determining the Differential Equation

Question

When a certain IV drug is being administered, it enters the bloodstream at a rate of 400 mg per hour. It also leaves the bloodstream at a rate proportional to the amount present. The rate constant is 0.8 (1/hour). Let Q(t) be the amount (in mg) of the drug in the bloodstream at time t (in hours). Write a differential equation for the function Q.

I know the general form is dQ/dt= 400+kQ, but I'm not entirely sure that is correct for this example. Similarly, I am struggling to find k.

Answer 1

It might be useful to consider first a differential way of writing the process.

At a certain point in time for a "differential" time interval $dt$ the equation for a "differential" change $dQ$ in $Q$ would look as follows:

$$dQ = 400 \cdot dt - 0.8\cdot Q \cdot dt $$

So, your $k$ is negative as the drug is leaving the body.

Answer 2

The equation is the linear $$ \frac {dQ}{dt} = 400-0.8Q(t)$$ and you can solve it by the method of integrating factor or even easier $$ Q=Q_c+ Q_p $$ Thus your $$ Q(t) = Ae^{-0.8t} +500$$ where you find $A$ from the initial value of $Q$