Perturbations of solutions of the Einstein equations with sources

Expressions for the complete perturbations of solutions of the Einstein-Maxwell equations and of the Einstein-Weyl equations in terms of potentials are derived, starting from decoupled sets of equations and using Wald's method (1979) of adjoint operators. The background electromagnetic field is required to be non-null with one of its principal null directions being geodetic and shear free and with an analogous requirement on the background neutrino field in the case of the Einstein-Weyl equations.