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Deriving the Bateman equation of Nuclear Decay Chains 
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#1
Mar1307, 08:31 PM

P: 1

1. The problem statement, all variables and given/known data
Derive Bateman equation for a decay chain a>b>c>d where each decays with a given mean life let decay constant be L, where L=1/mean life Na(0)=No, Nb(0)=Nc(0)=Nd(0)=0 2. Relevant equations Want to derive Nb(t)={(No)(La)/(LbLa)}*{exp[La*t]exp[Lb*t]} extend for Nc(t) 3. The attempt at a solution dNa(t)/dt=La*Na(t) Na(t)=No*exp[La*t] dNb(t)/dt=Lb*Nb(t)+LaNa(t) dNb(t)/dt=Lb*Nb(t)+La{No*exp[La*t]} this is a none homogenous differential equation. I can't find a way to solve it. dNc(t)/dt=Lc*Nc(t)+LbNb(t) I'm really not sure where to go from here. If anyone could lend a hand it would be greatly appreciated. 


#2
Mar1407, 11:09 AM

Sci Advisor
HW Helper
Thanks
P: 25,235

Ok, so what you are really asking is how to solve y'(t)+L*y(t)=f(t). You know the homogenous solution is exp(L*t). Guess the solution will be of the form y(t)=g(t)*exp(L*t). Put this guess into your original equation and get:
g'(t)*exp(L*t)L*g(t)*exp(L*t)+L*g(t)*exp(L*t)=f(t). So g'(t)=exp(L*t)*f(t) and you can just integrate to get g(t). 


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