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In this work , the Laplace Adomian-Decomposition method was used to solve Duffing oscillator equation. The solution to both integer and fractional case Duffing Oscillator was discussed in this work. Different types of the integer case Duffing oscillator was solved, and the analytic solution was gotten. The graphical plots also have been presented for different values of damping parameter. From the graph, we have given the conclusion for the nature of damping. The two-kind fractional Duffing was also considered, and analytical solutions/conclusion was drawn. Stability analysis of the linear and non-linear system was considered on both integer and fractional case. The work also considered the construction of Lyapunov function by Cartwright method for Duffing type equation discussed here.