Linear first-order diffeq system for radioactive decay chain

[clynne21]

Homework Statement

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)=$\alpha$t and assuming N_y(t)=N_z(t)=0

Homework Equations

dN_x(t)/dt = -$\lambda$_xN_x(t) + R_x(t)
dN_y(t)/dt = -$\lambda$_yN_y(t) +$\lambda$_xN_x(t)
dN_z(t)/dt = -$\lambda$_zN_z(t) +$\lambda$_yN_y(t)

The Attempt at a Solution

I know these would be solved with bateman equations and without the R_x(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 N_x(t) = N_x(0)e^{-$\lambda$_xt} + ∫^t₀ dt'R_x(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 R_x(t)=$\alpha$t integrate and where does it go in the other two equations? Thanks!

 

[HallsofIvy]

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.

 

[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.