Radioactive Decay Problem

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:

**1. The problem statement, all variables and given/known data**

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:

<b>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: (B(t): the amount of Bismuth at any time)

<b>dB/dt= -b.B + a.L

</b> (a.L: quantity of lead decayed added to the Bismuth which equals 100.e^(-.035185.t)) and b: constant of decay for Bismuth

Solving this 1st Order DE we get:

<b>[100.e^(-.035185.t) + C.100.e^(-b.t)]

B(t)= _____________________________________

</b>

C is a constant of integration.

First: Is my B(t) equation correct based on the problem we have?

And second, how can I solve for b and C based on the IVP I have?

Can you guys help?

thx :)