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Homework Help: Coupled Diff EQ

  1. Mar 3, 2010 #1
    1. The problem statement, all variables and given/known data
    Consider a radiactive decay problem involving two types of nuclei, A and B, with populations Na(t) and Nb(t). Suppose that type A nuclei decay to form type B nuclei, which then also decay, according to differential equations:

    2. Relevant equations

    dNa/dt = - Na/Ta

    dNb/dt = Na/Ta - Nb/Tb

    where Ta and Tb are the decay constants. Na(0) = 100 and Nb(0) = 0.
    3. The attempt at a solution

    I solved the equation for Na(t) and got 100*e^(-t/Ta), but I'm not sure about Nb. Intuitively, I think it has to be something like:

    100*(1 - e^(-t/Ta))*e^(-t/Tb)

    but I know this is wrong

    any advice/ideas?
  2. jcsd
  3. Mar 3, 2010 #2
  4. Mar 3, 2010 #3


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    Staff Emeritus
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    Just plug it in and solve! ;)

    Your equation for Nb,

    [tex]N'_b(t)+\frac{1}{T_b} N_b(t) = 100e^{-t/T_a}[/tex]

    is a basic first-order, inhomogeneous differential equation. You can solve it using an integrating factor or the method of undetermined coefficients.
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