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!