(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given the followin[Sg decay chain- X→Y→Z

Solve for N_{x}(t), N_{y}(t), N_{z}(t) for the case of R_{x}(t)=[itex]\alpha[/itex]t and assuming N_{y}(t)=N_{z}(t)=0

2. Relevant equations

dN_{x}(t)/dt = -[itex]\lambda[/itex]_{x}N_{x}(t) + R_{x}(t)

dN_{y}(t)/dt = -[itex]\lambda[/itex]_{y}N_{y}(t) +[itex]\lambda[/itex]_{x}N_{x}(t)

dN_{z}(t)/dt = -[itex]\lambda[/itex]_{z}N_{z}(t) +[itex]\lambda[/itex]_{y}N_{y}(t)

3. The attempt at a solution

I know these would be solved with bateman equations and without the R_{x}(t)=[itex]\alpha[/itex]t term I could do these. The production term throws me off and I'm not sure exactly how to go about this.

I have this for N_{x}(t) = N_{x(0)}e^{-[itex]\lambda[/itex]xt}+ ∫^{t}_{0}dt'R_{x}(t')e^{[itex]\lambda[/itex]x(t'-t)}(the integral is from 0 to t, but the itex wasn't working for me to do that)

So how does R_{x}(t)=[itex]\alpha[/itex]t integrate and where does it go in the other two equations? Thanks!

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# Linear first-order diffeq system for radioactive decay chain

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