The Dirichlet Boundary Value Problem for Real Solutions of the first   Painlev\'e Equation on Segments on Non-Positive Semi-Axis
  We develop a qualitative theory for real solutions of the equation $y''=6y^2 -x$. In this work a restriction $x\leq0$ is assumed. An important ingredient of our theory is the introduction of several new transcendental functions of one, two, and three variables that describe different properties of the solutions. In particular, the results obtained allow us to completely analyse the Dirichlet boundary value problem $y(a)=y^0$, $y(b)=y_0$ for $a<b\leq0$. 
N. Joshi and A. V. Kitaev