(adsbygoogle = window.adsbygoogle || []).push({}); 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)/(Lb-La)}*{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.

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# Deriving the Bateman equation of Nuclear Decay Chains

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