Nilpotent Algebraic Representation of Linear Systems with Polynomial–Exponential Forcing: Theory and Application to Coupled RLC Circuits

We develop an exact operational solution for second-order linear differential systems associated with coupled two-mesh RLC circuits subject to polynomial-exponential forcing. The method is based on the Nilpotent Operational System (SON), where the derivative, restricted to finite-dimensional polynomial spaces, is represented by a nilpotent matrix. This transforms the search for particular solutions into a finite linear algebra problem. The nonresonant and resonant cases are both treated. The method is shown to be equivalent to the classical method of undetermined coefficients, while providing a compact, structural, and extensible matrix formulation. The homogeneous solution for the example is computed in an appendix via classical eigenvalue methods.