<p>This paper is concerned with the pressureless Euler equations with source terms, which can model acoustic generation in a fluid by forcing. With the use of the substitution of variables, two kinds of non-self-similar Riemann solutions are constructed: vacuum solution and delta-shock solution. The generalized Rankine–Hugoniot relation for the delta-shock is clarified. Under delta-entropy condition, both existence and uniqueness of the delta-shock solution are proved by the generalized Rankine–Hugoniot relation. The interesting discovery is that even if the trajectory of the delta-shock is a straight line in the characteristic plane, its weight is no longer the linear function of time. Moreover, the Riemann problem for the balance equations with delta initial data is solved. Four different kinds of structures of solutions are constructed. Additionally, with the vanishing viscosity method, the stability of the non-self-similar solutions is established by introducing a time-dependent viscous system.</p>