Summary: Many problems in science and engineering fields can be described by differential equations. In the early 1980's, an American applied mathematician George Adomian developed a powerful decomposition methodology for practical solution of differential equations known today as the Adomian decomposition method (ADM). The ADM is a powerful method which provides an efficient means for the analytical and numerical solution of differential equations which model real-world physical problems. The differential transform method (DTM) was first proposed by Zhou in 1986. The DTM is used to find coefficients of the Taylor series of the function by solving the induced recursive equation from the given differential equation. Recently there has been a big debate among researchers on which method is the best method to solve nonlinear differential equations. The DTM is clearly documented and well understood for solving ordinary differential equations. In this paper, we apply the ADM and clearly document how the DTM can be used to solve both ordinary differential equations (ODE's) and partial differential equations (PDE's).