For the barotropic fluid, based on the quasi-geostrophic potential vorticity equation, an inhomogeneous Boussinesq equation including topographic forcing and an external source is derived by employing the perturbation method and stretching transform of time and space. Through inspection of the evolution of the amplitude of the Rossby wave, it is found that Coridis effect, topography effect and an external source are the key inducing factors of the solitary Rossby wave if the basic stream function has a shear flow. Assuming that there is a balance between nonlinear and topography effects, an inhomogeneous Boussinesq equation is derived. The results show that the topography and the Rossby wave interact in the barotropic flow. The inhomogeneous Boussinesq equation describing the evolution of the amplitude of solitary Rossby waves with the change of Rossby parameter β(y) with latitude β, topographic forcing and the external source is obtained.