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baouba
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Homework Statement
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
Homework Equations
My Prof said Nb(t) has the form Nb(t) = Cexp(-t/Ta) + Dexp(-t/Tb)
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