Construction of Solutions of Gravitational, Electromagnetic, or Other Perturbation Equations from Solutions of Decoupled Equations

A simple proof is given of a general result which shows how to construct solutions of a coupled system of linear self-adjoint partial differential equations once one has succeeded in deriving a decoupled equation in the manner described below. In the case of electromagnetic and gravitational perturbations of algebraically special vacuum space-times, this procedure yields the formulas of Cohen and Kegeles and of Chrzanowski.