# Linear first-order diffeq system for radioactive decay chain

1. Jan 26, 2012

### clynne21

1. The problem statement, all variables and given/known data
Given the followin[Sg decay chain- X→Y→Z
Solve for Nx(t), Ny(t), Nz(t) for the case of Rx(t)=$\alpha$t and assuming Ny(t)=Nz(t)=0

2. Relevant equations
dNx(t)/dt = -$\lambda$xNx(t) + Rx(t)
dNy(t)/dt = -$\lambda$yNy(t) +$\lambda$xNx(t)
dNz(t)/dt = -$\lambda$zNz(t) +$\lambda$yNy(t)

3. The attempt at a solution
I know these would be solved with bateman equations and without the Rx(t)=$\alpha$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 Nx(t) = Nx(0)e-$\lambda$xt + ∫t0 dt'Rx(t')e$\lambda$x(t'-t) (the integral is from 0 to t, but the itex wasn't working for me to do that)
So how does Rx(t)=$\alpha$t integrate and where does it go in the other two equations? Thanks!

Last edited: Jan 26, 2012
2. Jan 26, 2012

### HallsofIvy

Staff Emeritus
Excuse me but are you saying that you are trying to solve a system of differential equations but do not know how to integrate $\alpha t$? The integral of $\alpha t$ with respect to t is $\alpha t^2/2$. That is usually one of the first integrals you learn.

Last edited: Jan 26, 2012
3. Jan 26, 2012

### clynne21

Haha- no, that's cake LOL

It's really more this term ∫dt'Rx(t')eλx(t'-t) (from 0 to t) that confuses me- I'm not sure where the primes came from and what it is indicating. It was the first step given for a solution. It seemed odd since the αt should be, like you said, extremely simple.