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Math Help - differential equation for amount of drug administered intravenously

  1. #1
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    differential equation for amount of drug administered intravenously

    A drug is administered intravenously at a constant rate of r mg/hour and is excreted at a rate proportional to the quantity present, with constant of proportionality k>0.

    (Set up and) Solve a differential equation for the quantity, Q, in milligrams, of the drug in the body at time t hours. Assume there is no drug in the body initially. Your answer will contain r and k.

    >>>

    I started with dQ/dt = r - kQ
    then got:
    (dQ/dt) / [(r/k) - Q] = k (I factored out k from the above equation)
    integrating:
    ln((r/k)-Q) = kt + C
    solving for Q, I get Q = (r/k) - ce^(kt)

    Where am I wrong? Thanks
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  2. #2
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    You've dropped a negative sign when you integrated the LHS.
    It should be -\ln(r/k-Q) .
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  3. #3
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    thanks

    so i get 1/[ (r/k) - Q] = ce^(kt)
    solving for Q, I get Q = (r/k) - e^(-kt) but this is still incorrect. any ideas?
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  4. #4
    MHF Contributor Calculus26's Avatar
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    Lets back up to

    ln(r/k -Q) = -kt + c

    r/k -Q = c e^(-kt)

    Apply the initial condition Q(0) = 0

    r/k = c

    Q= r/k - r/k e^(-kt)
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  5. #5
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    thank you so much, that was a silly mistake
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