  We develop a formally exact technique for obtaining steady-state
distributions of non-interacting active Brownian particles in a variety of
systems. Our technique draws on results from the theory of two-way diffusion
equations to solve the steady-state Smoluchowski equation for the 1-particle
distribution function. The methods are employed to study in detail three
scenarios: 1) confinement in a channel, 2) a constant flux steady state, and 3)
sedimentation in a uniform external field. In each scenario, known behaviors
are reproduced and precisely quantified, and new results are presented. In
particular, in the constant flux state we derive an effective diffusivity which
interpolates between the ballistic behavior of particle trajectories at short
distances and their diffusive behavior at large distances. We also calculate
the sedimentation profile of active Brownian particles near a wall, which
complements earlier studies on the part far from the wall. Our techniques
easily generalize to other active models, including systems whose activity is
modeled in terms of Gaussian colored noise.
