2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72001We 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 $a<b\leq0$.To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal)Classical Analysis and ODEsMathematical PhysicsExactly Solvable and Integrable Systems33E17;34M55;34B15text