We generalize the Newton polygon procedure for algebraic equations to generate solutions of polynomial differential equations of the form ∑ i = 0 ∞ α i x β i \sum \nolimits _{i = 0}^\infty {{\alpha _i}{x^{{\beta _i}}}} where the α i {\alpha _i} are complex numbers and the β i {\beta _i} are real numbers with β 0 > β 1 > ⋯ {\beta _0} > {\beta _1} > \cdots . Using the differential version of the Newton polygon process, we show that any such a series solution is finitely determined and show how one can enumerate all such solutions of a given polynomial differential equation. We also show that the question of deciding if a system of polynomial differential equations has such a power series solution is undecidable.