I've been working with this problem for almost two weeks trying
to find a good equation for the decay of Bismuth to no avail.
Can someone give me insightful comments:  Here's the problem:
In the radioactive decay series of Uranium (238, 92), isotopes
of lead, bismuth occur as products of two successive Beta decays
with half -lives of 19.7 minutes and 26.8 minutes, respectively.
U --> Lead --> Bismuth  Decays are each proportional to the
amount of isotope present. Assume initially at time =0, we have
100 mg of lead and 150 mg of bismuth?  We are asked to find the
amount of lead and bismuth at any time?  2. Relevant equations
So first we have to formulate a DE for the decay.  The one
equation for Lead is simple.  Let L(t) be the amount of lead at
any time, then the DE model is:  dL/dt = -a.L </b> (a: constant
of decay)  after integration and get L(t) = 100.e^(-a.t)  Using
the Initial Value Problem and half-life value we get an
equation: L(t)= 100.e^(-.035185.t)  Now the Differential
Equation for Bismuth is