Differential Equations and drugs

1. Aug 24, 2010

kukumaluboy

Let x mg be the amount of drug in the patient at time t.

Drug is injected into the patient at a rate of P mg per min

The kidneys of the patient remove the drug at a rate proportional to the amount of drug at time t.

At a particular point in time, the drug concentration in the patient remains constant. This constant value is 2.5 P mg

Show that dx/dt = P - 0.4x

Hence express x in terms of P and t

2. Aug 24, 2010

vela

Staff Emeritus
You need to show some attempt at the problem before you'll get any help.

3. Aug 24, 2010

kukumaluboy

Alrite

Let x mg be the amount of drug in the patient at time t.

dx/dt = P + Kx, where P>0 and K<0

Solving by Integrating Factor;

dx/dt -Kx =P

P(t) = -K;
Integrate P(t) wrt t = -Kt;
Hence integrating Factor = e^(-Kt);

xe^(-Kt) = P * Integrate(e^(-Kt)) dt;
xe^(-Kt) =(-P/K) * e^(-Kt) + C;

At a particular point in time, the drug concentration in the patient remains constant. This constant value is 2.5 P mg
Means P and K are equal. Henve -P/K at that time will be -1
Then dunno how to do

4. Aug 24, 2010

vela

Staff Emeritus
One comment: It's not a good idea to use P(t) in your notation for the integrating factor since you already use P to stand for something else in the problem.

If you solve for x(t), you get x(t) = (-P/K) + C eKt.

Second comment: Your work so far is correct, but it's traditional to take K>0 and to put the negative sign into the differential equation explicitly. If you do that, your answer would come out to be x(t) = (P/K) + C e-Kt.
When it says the concentration remains constant, that does not mean -P/K=-1. Remember, the drug concentration is given by x(t), so to say it remains constant is a statement about x(t) and dx/dt.

5. Aug 24, 2010

kukumaluboy

Oh yea!
Well Said.

Re-did the qn and got x(t) = (P/K) + C e-Kt like you.

At a particular point in time, the drug concentration in the patient remains constant. This constant value is 2.5 P mg

So this actually means :

dx/dt = P-Kx = 0 at that time
P=Kx
x was 2.5P
P=2.5PK
K=0.4
Hence dx/dt = P-0.4x

Thanks the rest are easy.