We consider a network with N nodes competing for access to the channel using un-slotted ALOHA. When a request is sent, each node may answer after a certain backoff time. Only the first answer is of importance, that is why we want to minimize the loss rate of the first message. We derive the optimal backoff probability distribution which minimizes the collision probability on the first message answering to a request. Unlike previous works, we extract the collision probability in continuous time domain. To this goal, we use a variational method. This problem had only be solved before in a slotted context (i.e. discrete time domain), but we want to be able to manage later situations where the nodes are not perfectly synchronized, which requires to know how to solve it in the continuous time context