Flat expansions and their applications

Polynomial ordinary differential equations (ODEs) near a singular point are considered. Families of solutions to ODEs that are exponentially close to a solution represented by a formal power series are studied. It is shown that, for systems of ODEs in the plane, all solutions of such a family are uniquely determined by a series of flat functions. Flat expansions are poorly understood. The power series involved in flat expansions can converge or diverge. Examples of computations of flat expansions are given, and their applications are considered. A flat expansion of the solution to the Blasius problem at infinity is calculated. It is shown that this asymptotic expansion can be matched with the Blasius power series expansion at the origin.