Systems that are below prompt critical are considered, and the linear time-dependent neutron transport equation in a quite general setting is studied. Both source and cross sections are allowed to depend on space, energy, and time. The method of matched asymptotic expansions is used to find an asymptotic solution uniformly valid in time. This solution is written in the form of a sum of solutions to simpler problems and for most practical problems is essentially exact. After a short initial time period, the transport equation (with delayed neutrons neglected) may be solved at a given time by a single inversion of the steady-state transport operator; i.e., with a steady-state code.