Linear birth-death process waiting time problem

Mar 2006
705
7
Suppose that \(\displaystyle X(t)\) is a linear birth-death process with \(\displaystyle \lambda = \mu \)

Let \(\displaystyle u(t)=P[X(t)=0 | X(0)=1 ] \)

Show that \(\displaystyle u(t)= \frac {1}{2} \int ^t_0 2 \lambda \exp (-2 \lambda s)ds + \int ^t_0 2 \lambda \exp (-2 \lambda s)(u(t-s))^2ds \)

Proof so far:

So I know that the waiting time until extinction has an exponential distrbution with rate of \(\displaystyle 2 \lambda \)...

So \(\displaystyle P[X(t)=0] = e^{-2 \lambda t} \), but how would I incorporate the other stuff in there? Thanks!