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

We consider ordinary differential equations (ODE) of the form $u^{\mathrm{\prime }\mathrm{\prime }}u-(u^{\mathrm{\prime }}{)}^2=e^{-x}P(u)-1$, 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 $u(x)={\sum }_{k=0}^{\mathrm{\infty }}{p}_k(x+c)e^{-kx}$ for $Rex\to +\mathrm{\infty }$, 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 $P^{\mathrm{\prime }}(0)=\frac{1}{2}{P}^{\mathrm{\prime }\mathrm{\prime }}(0)$. Here we show that these two conditions are also sufficient. The results imply the existence of corresponding expansions for certain degenerate Painlevé III transcendents.