In the past, theorems have shown that individuals can implement a (formal) power series method to derive solutions to Algebraic Ordinary Differential Equations, or AODEs. First, this paper will give a quick synopsis of these “bottom-up” approaches while further elaborating on a recent theorem that established the (modified) Generating Function Technique, or mGFT, as a powerful method for solving differentials equations. Instead of building a (formal) power series, the latter method uses a predefined set of (truncated) Laurent series comprised of polynomial linear, exponential, hypergeometric, or hybrid rings to produce an analytic solution. Next, this study will utilize the mGFT to create analytic solutions to a several example AODEs.Ultimately, one will find mGFT may serve as a powerful "top-down" method for solving linear and nonlinear AODEs.
