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How do I solve these coupled Differential Equation?

  1. Sep 29, 2015 #1
    1. The problem statement, all variables and given/known data
    dNa/dt = -Na/Ta where Na is the function and Ta is the constant
    dNb/dt = Na/Ta - Nb/Tb where Nb is the function and Tb is the constant

    2. Relevant equations
    My Prof said Nb(t) has the form Nb(t) = Cexp(-t/Ta) + Dexp(-t/Tb)

    3. The attempt at a solution
    I know the first equation solves to Na(t) = Na(0)exp(-t/Ta)

    The second equation can be written,

    dNb/dt = (TbNa-TaNb)/(TaTb)

    (TaTb)dNb/dt = TbNa-TaNb

    Separation of variables and integrating gives:

    (TaTb) ∫ [TbNa-TaNb]^-1 dNb = ∫dt

    (TaTb)(-1/Ta)ln(NaTb - NbTa) = t + C

    Rearranging,

    Nb = (Tb/Ta)Na - Cexp(-t/Tb)

    Subbing in Na,

    Nb = (Tb/Ta)(Na(0)exp(-t/Ta)) - Cexp(-t/Tb)

    at t = 0,

    Nb(0) = (Tb/Ta)(Na(0)) - C

    so C = (Tb/Ta)(Na(0)) - Nb(0)

    Subbing back into Nb(t),

    Nb = [(Tb/Ta)(Na(0)exp(-t/Ta))] - [(Tb/Ta)(Na(0)) - Nb(0)]exp(-t/Tb)

    Apparently the right answer is,

    Nb = (1 / ((Ta/Tb)-1.)) Na(0)exp(-t/Ta)+ (Nb(0)-(Na(0)/((Ta/Tb)-1.)))exp(-t/Tb);

    but I just can't seem to get it. Can anyone tell me where I went wrong?

    Thanks
     
  2. jcsd
  3. Sep 29, 2015 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Once you have solved the first equation for ##N_a(t)##, your second equation becoms
    [tex] \frac{dN_b}{dt} = f(t) - \tau_b N_b, [/tex]
    where ##\tau_b \equiv 1 / T_b## is a constant and ##f(t) = N_a(t)/T_a## is a known function. You can solve this DE using an integrating-factor approach; see, eg.,
    https://en.wikipedia.org/wiki/Integrating_factor .
     
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