With the use of the operation of integrating the divergent series of a formal solution in the separating variables method, there are obtained the results concerning a generalized mixed problem (homogeneous and non-homogeneous) for the wave equation. The key moment consists in finding the sum of the divergent series  which corresponds to the simplest mixed problem with a summable initial function. This result helps to get  the solution of the generalized mixed problem for a non-homogeneous equation under the assumption that non-homogeneity is characterized by a locally summable function. As an application, the mixed problem with a non-zero potential is considered, in which the differential equation is treated quite formally but the  mixed problem itself is no longer a generalized one: instead of the formal solution of the separating  variables method we get an integral equation which can be solved by the successive substitutions method. Thus we essentially simplify the arguments.