  We derive mean-field equations for a general class of ferromagnetic spin
systems with an explicit error bound in finite volumes. The proof is based on a
link between the mean-field equation and the free convolution formalism of
random matrix theory, which we exploit in terms of a dynamical method. We
present three sample applications of our results to Ka\'{c} interactions,
randomly diluted models, and models with an asymptotically vanishing external
field.
