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Coupled Diff EQ

  • #1

Homework Statement


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:


Homework 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.

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?
 

Answers and Replies

  • #2
.......anyone?
 
  • #3
vela
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Homework Statement


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:


Homework 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.

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?
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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