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Math Help - Differential Equations in clinical medicine problem

  1. #1
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    Differential Equations in clinical medicine problem

    The question is solve R(dVi/dt) + (1/C)Vi + Pex = Papp , 0 t ti


    where

    Vi (0) = 0

    i, ex, app are subscripts
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  2. #2
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    Quote Originally Posted by targus2 View Post
    The question is solve R(dVi/dt) + (1/C)Vi + Pex = Papp , 0 t ti


    where

    Vi (0) = 0

    i, ex, app are subscripts
    What is R, C, P_{ex} \;\text{and}\; P_{app} ? Are they constants?
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    yes
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  4. #4
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    So

     <br />
\frac{d V_i}{dt} + \frac{V_i}{RC} = \frac{P_{app} - P_{ex}}{R}<br />

    The integrating factor is e^{t/RC} so this gives

     <br />
\frac{d}{dt} V_i e^{t/RC} = \frac{P_{app} - P_{ex}}{R} e^{t/RC}<br />

    Integrating gives

    V_i e^{t/RC} = C \left(P_{app} - P_{ex}\right) e^{t/RC} + V_0

    or

     <br />
V_i = C\left(P_{app} - P_{ex}\right) + V_0 e^{-t/RC}.<br />

    Imposing the initial condition gives V_0 = - C\left(P_{app} - P_{ex}\right)

    and thus the final solution

    V_i = C \left(P_{app} - P_{ex}\right) \left(1 - e^{t/RC} \right)
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