We derive a nonlinear second-order differential equation for the pressure approximation in hydrodynamic lubrication. This equation, in contrast to the classical Reynolds equation, takes into account both the inertial and the curvature effects and its solution corresponds to the first two terms in the asymptotic pressure expansion. The equation is rigorously justified with optimal error estimates in parameter dependent Sobolev norms and in Hölder norms. It is also applied to the classical problem of journal bearing.