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Homework Help: Differential Equations and drugs

  1. Aug 24, 2010 #1
    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. jcsd
  3. Aug 24, 2010 #2

    vela

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    You need to show some attempt at the problem before you'll get any help.
     
  4. Aug 24, 2010 #3
    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
     
  5. Aug 24, 2010 #4

    vela

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    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.
     
  6. Aug 24, 2010 #5
    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.
     
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