One-third-power scaling at a nonlinear turning endpoint ($x={ϵ}^{1/3}X,y={ϵ}^{1/3}Y$)

In $ϵy^{\prime \prime }+xy+y^2=0$, balancing all three terms where the outer branches $0$ and $-x$ meet gives the displayed dominant balance and the inner equation $Y^{\prime \prime }+XY+Y^2=0$. Matching to $-X$ and writing $Y=-X+u$ gives $u^{\prime \prime }-Xu+u^2=0$. Its linearized decaying correction is an Airy function; the growing correction is excluded. Matching to zero instead gives $Y^{\prime \prime }+XY=0$, whose nonzero Airy function combinations oscillate. Thus the acceptable matching branches differ qualitatively at the turning endpoint.