## Linear birth-death process waiting time problem

Suppose that $X(t)$ is a linear birth-death process with $\lambda = \mu$

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

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

Proof so far:

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

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