This paper presents an integral-equation formulation specialized for the analysis of capacitive waveguide circuits, which include arbitrarily shaped conducting and homogeneous magnetic/dielectric objects. The technique benefits from the symmetry of the structure by formulating a 2-D scattering problem with oblique angle of incidence, combined with the use of the parallel-plate Green's functions. As practical applications, the paper proposes novel low-pass filter designs loaded or coated with dielectric and magnetic homogeneous materials. If the filter is properly designed, the use of these materials could improve the filter response, selectivity, or out-of-band performance, and power-handling capabilities. Some novel filter design implementations, showing these kinds of benefits, are presented for the first time in this paper. For validation, a commercial full-wave simulator is employed, showing the validity, accuracy, and computational efficiency of the novel software tool.