The paper provides a detailed study of the Adomian decomposition method (ADM) and the Picard method (PM) for solving ordinary differential equations (ODEs) related to electric circuits, specifically focusing on RC and RL circuits. It clearly establishes the existence and uniqueness of solution, while exploring how the series solutions converge and conducting a careful error analysis. This examination not only strengthens the theoretical understanding of these methods but also offers useful insights into their practical applications in electrical engineering and circuit analysis.