  We present a new approach to solving polynomial ordinary differential
equations by transforming them to linear functional equations and then solving
the linear functional equations. We will focus most of our attention upon the
first-order Abel differential equation with two nonlinear terms in order to
demonstrate in as much detail as possible the computations necessary for a
complete solution. We mention in our section on further developments that the
basic transformation idea can be generalized to apply to differential equations
of any order, to a system of ordinary differential equations without first
differentially eliminating the multiple dependent variables, and even to
partial differential equations.
