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Math Help - ODE Word Problem

  1. #1
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    ODE Word Problem

    Drug Metabolism The rate at which a drug is absorbed into the bloodstream is modeled by the first-order differential equation:



    where a and b are positive constants and C(t) denotes the concentration of the drug in the bloodstream at time . Assume no drug is initially present in the bloodstream.

    Find a formula for C(t). You may need to use a and b in your answer.


    I think it may be the constants a and b, but something's throwing me off on how to do this. So far I've tried:

    dC/dt = a-bC(t)
    dC/dt +C(t)= a

    And then I'm guessing I need to use an integrating factor? Or am I just completely wrong here. Any help is greatly appreciated!
    Last edited by cdlegendary; April 16th 2010 at 02:49 PM.
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  2. #2
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    Quote Originally Posted by cdlegendary View Post
    Drug Metabolism The rate at which a drug is absorbed into the bloodstream is modeled by the first-order differential equation:



    where a and b are positive constants and C(t) denotes the concentration of the drug in the bloodstream at time . Assume no drug is initially present in the bloodstream.

    Find a formula for C(t). You may need to use a and b in your answer.


    I think it may be the constants a and b, but something's throwing me off on how to do this. So far I've tried:

    dC/dt = a-bC(t)
    dC/dt -C(t)= a

    And then I'm guessing I need to use an integrating factor? Or am I just completely wrong here. Any help is greatly appreciated!
    \frac{dC}{dt} = a - b\,C

    \frac{dC}{dt} + b\,C = a.

    This is first order linear, so use the integrating factor e^{\int{b\,dt}} = e^{b\,t}.

    Multiplying through gives

    e^{b\,t}\frac{dC}{dt} + b\,e^{b\,t}C = a\,e^{b\,t}

    \frac{d}{dt}(C\,e^{b\,t}) = a\,e^{b\,t}

    C\,e^{b\,t} = \int{a\,e^{b\,t}\,dt}

    C\,e^{b\,t} = \frac{a\,e^{b\,t}}{b} + d

    C = \frac{a}{b} + d\,e^{-b\,t}.


    When t = 0, C = 0.

    So 0 = \frac{a}{b} + d\,e^{0}

    0 = \frac{a}{b} + d

    d = -\frac{a}{b}.


    Therefore

    C = \frac{a}{b} - \frac{a}{b}\,e^{-b\,t}.
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  3. #3
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    How could you find the limiting concentration as t approaches infinity? and at what time does it reach its half life?

    I can't do this problem!!! =[
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