  To study the depinning transition in the limit of strong disorder, Derrida
and Retaux (2014) introduced a discrete-time max-type recursive model. It is
believed that for a large class of recursive models, including Derrida and
Retaux' model, there is a highly non-trivial phase transition. In this article,
we present a continuous-time version of Derrida and Retaux model, built on a
Yule tree, which yields an exactly solvable model belonging to this
universality class. The integrability of this model allows us to study in
details the phase transition near criticality and can be used to confirm the
infinite order phase transition predicted by physicists. We also study the
scaling limit of this model at criticality, which we believe to be universal.
