Exact Operational Solution of RLC Circuits with Polynomial-Exponential Forcing via Nilpotent Calculus

We present an exact algebraic method for solving RLC circuits driven by polynomial-exponential forcing signals V(t) = P(t)eᵅᵗ, where P(t) is a polynomial of degree n. The method is based on the nilpotency of the differentiation operator D = d/dt on the finite-dimensional space Vₙ of polynomials of degree at most n. Since Dⁿ⁺¹ = 0 exactly in Vₙ, the Neumann series associated to the operator inverse p(D)⁻¹ truncates to a finite sum, yielding closed-form particular solutions without recourse to the Laplace transform. The method handles RL, RC, RLC series, RLC parallel, and higher-order circuits in a unified framework. Complete solutions with initial conditions, the resonant case, and five worked physical examples are included. Equivalence with undetermined coefficients and the Laplace transform is proved via Propositions 10.1 and 10.2.