Obtaining　PDE　backstepping　controller　or　observer　gains　requires　the　solution　of　kernel　PDEs　-　one　or　more　hyperbolic　PDEs　on　a　triangular　(Goursat)　domain　with　non-standard　boundary　conditions.　The　numerical　solution　of　these　equations　is　a　challenge　that　every　designer　applying　backstepping　must　eventually　face,　except　for　the　simplest　of　cases　where　explicit　solutions　are　available.　In　addition,　recent　backstepping　designs　for　coupled　systems　exhibit　discontinuous　behavior　which　must　be　accurately　captured　with　the　numerical　approximation.　In　this　paper,　we　propose　a　power　series　method　as　an　alternative　to　other　approaches.　This　method,　which　was　introduced　years　ago　in　combination　with　convex　optimization　but　whose　convergence　has　only　been　recently　formally　established,　offers　several　advantages,　most　of　all　simplicity,　as　it　is　quite　easy　to　grasp　and　implement.　Other　features　of　the　method　include　precision,　adaptability　to　settings　with　discontinuous　kernels,　and　the　ability　to　produce　symbolical　kernels　depending　on　parameters.　The　paper　provides　the　necessary　theoretical　background,　which　borrows　fundamental　results　from　complex　analysis　and　leverages　already　written　kernel　well-posedness　proofs　to　show　the　existence　of　a　power　series　solution.　Links　to　the　codes　used　in　the　simple　examples　are　given;　these　are　easily　adaptable　Mathematica　notebooks.　A　complex　multi-kernel,　multiple-discontinuity　example　used　in　the　stabilizing　feedback　law　of　a　multilayer　Timoshenko　beam　is　given　at　the　end,　demonstrating　the　applicability　of　the　method　to　some　challenging　families　of　kernel　equations　appearing　in　recent　backstepping　designs.