A mathematical model for the diffusion of dust particles emitted from a fixed source is investigated using the atmospheric diffusion equation. This model poses an initial boundary value problem with a second order linear partial differential equation. The steady state case of this problem when the uniform source is situated at ground level was examined by Sharan et al. [1]. The solution of the unsteady case in closed form for a time dependent source is derived. Two special cases, in which the source function of time is explicitly given and special values of the diffusion parameters are taken, are examined in detail. In the case when diffusion is present only in the vertical direction, it is shown that for small times, the particles spread with a front that travels with the speed of the wind. When diffusion is present only in the direction of the wind, there is no discontinuity front and the particles diffuse slowly into the direction of the wind. The solutions for the special cases considered are examined for large values of time. It is found that the solution approaches that of the corresponding steady state solution of the equation.