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Adomian s decomposition method is a useful and powerful method for solving linear and nonlinear ordinary differential equations. In this book we consider singular initial value problems, singular boundary value problems, boundary value problems and nonlinear oscillatory equations. Singular initial-value problems in second-order ordinary differential equations of Lane-Emden type are investigated.In this study, we shall introduce a further development in the Adomian decomposition method to overcome the difficulty at linear and non-linear Lane-Emden-like equations; especially when the singularity appears two times on the left -hand side of this type of equations. A more general concept of this type of initial-value problems is introduced with various examples. The numerical results of these examples are compared with exact solutions when available. A fast and accurate algorithm is developed for the solution of higher order boundary value problems with boundary conditions. A modifiedform of the Adomian decomposition method is applied to construct the numerical solution for such problems.