Starting with a linear potential of the form  the corresponding Smoluchowski equation becomes,
Where the diffusion constant, , is constant over space and time. The boundary conditions are such that the probability vanishes at  with an initial condition of the ensemble of particles starting in the same place, .
Defining  and  and applying the coordinate transformation,
With  the Smoluchowki equation becomes,
Which is the free diffusion equation with solution,
And after transforming back to the original coordinates,