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!