Necessary Conditions for the Existence of Exponential-Polynomial Expansions for Solutions of Certain Differential Equations

We consider ordinary differential equations (ODE) of the form u′′u−(u′)2=e−xP(u)−1, where P is a polynomial. For P=uk, k=3,4,6 this ODE is equivalent to certain degenerate Painlevé III equations. We study whether families of solutions of these ODEs have asymptotic expansions of the form u(x)=∑k=0∞pk(x+c)e−kx for Rex→+∞, where c∈C is an arbitrary constant parameterizing the solution family, pk are polynomials, with p0(x)=x. We find necessary conditions on P for such expansions to exist. Numerical experiments suggest that these conditions are also sufficient, and the expansions are not only formal, but actually provide a series representation of the solutions. Numerical evidence also suggests a conjecture on the nonnegativity of coefficients of the pk.