I am trying to answer the question
Species A decays at rate α into species B and species B decays at rate β into species C.
Show that if $ a \neq b $, then...
$ A(t) = A_0e^{-\alpha t}, \quad B(t) = B_0e^{-\beta t} + \alpha A_0(\frac{e^{-\alpha t}-e^{-\beta t}}{\beta - \alpha}) $
The first part I found extremely easy, however the second part I can't see where I'm going wrong, perhaps because I am extremely tired! :P
Here's my working so far:
$ \frac{dB}{dt} = \alpha A - \beta B \rightarrow \int\frac{1}{\alpha A - \beta B} dB = \int dt \rightarrow -\frac{1}{\beta} ln(\alpha A - \beta B) = t+c \rightarrow \alpha A - \beta B = e^{-\beta (t+c)}$
Setting t=0 we get:
$ \alpha A_0 - \beta B_0 = e^{-\beta c} \rightarrow c = \frac{ln(\alpha A_0 - \beta B_0)}{-\beta} $
Subbing in c, I then get the equation to be:
$B\beta = \alpha A - e^{-\beta t}e^{ln(\alpha A_0 - \beta B_0))}$
Which then along with the equation for A simplifies to
$B = \frac{\alpha A_0}{\beta}(e^{-\alpha t}-e^{-\beta t}) + B_0e^{-\beta t}$
So the only thing I cannot get right is the $\beta - \alpha$ on the denominator... I must be making a silly error somewhere, but I can't see where! Can anyone point me in the right direction? I don't want a full work-through, just pointers please :D