This paper introduces a novel approach that integrates federated optimization, Extreme Learning Machine (ELM), and pure differential privacy, tailored specifically for edge computing scenarios. The proposed algorithm, Differentially Private Federated Extreme Learning Machine (DP-FedELM), ensures data privacy while enabling collaborative training across distributed trainers with their local datasets. In DP-FedELM, trainers do not share raw data; instead, they exchange statistical values protected by pure differential privacy, governed by a privacy budget \(\epsilon \) . We establish the theoretical foundation of DP-FedELM and demonstrate its practical effectiveness through experiments conducted with reasonable values of the privacy budget \(\epsilon \) .