Equation of Motion and Energy Analysis of Nonlinear Systems using MDTM

We apply the Modified Differential Transformation Method (MDTM) to obtain semi-analytic series solutions of second-order equations of motion, including nonlinear terms and forcing/damping. MDTM produces a rapidly convergent recurrence for the power-series coefficients of the solution; nonlinearities are handled via discrete convolution formulae. We show how to derive an exact energy balance (time derivative of mechanical energy) from the equation of motion and how to compute the series expansion of the energy using the MDTM coefficients. A worked example (Duffing oscillator with viscous damping) illustrates the method and gives explicit first terms of the solution and energy series. The study includes an algorithmic description suitable for numerical implementation. Keywords – MDTM; DTM; Equation of Motion; Equation of Energy; non-linear; ODF.