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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