Tritronquée Solutions of Perturbed First Painlevé Equations

We consider solutions of the class of ODEs $y''=6y^2-x^{\mu}$, which contains the first Painlevé equation $($PI$)$ for $\mu=1$. It is well known that PI has a unique real solution (called a tritronquйe solution) asymptotic to $-\sqrt{x/6}$ and decaying monotonically on the positive real line. We prove the existence and uniqueness of a corresponding solution for each real nonnegative $\mu\ne1$.