Sufficient conditions for the existence of exponential-polynomial expansions for solutions of certain differential equations

Abstract

We consider ordinary differential equations (ODE) of the form $u''u - (u')^2 = e^{-x}P(u) - 1$, where $P$ is a polynomial. In previous work, necessary conditions on $P$ have been established for certain families of solutions of these ODEs to have asymptotic expansions of the form $u(x) = \sum_{k=0}^{\infty} p_k(x+c)e^{-kx}$ for $Re\,x \to +\infty$, where $c \in \mathbb C$ is an arbitrary constant parameterizing the solution family, and $p_k$ are polynomials, with $p_0(x) = x$. These conditions amount to $P(0) = 0$ and $P'(0) = \frac12P''(0)$. Here we show that these two conditions are also sufficient. The results imply the existence of corresponding expansions for certain degenerate Painlev\'e III transcendents.