= Nonlinear endpoint layer with a quadratic reaction
{title2=$Y''+qY+Y^2=0$}
For $q>0$, matching $Y\to-q$, $Y'\to0$ fixes the <first integral> to $(Y')^2=(q-2Y)(Y+q)^2/3$. Hence endpoint values above $q/2$ cannot match this outer branch. The upper family is $Y=-q+(3q/2)\operatorname{sech}^2[\sqrt q(\xi-\xi_0)/2]$. An endpoint value strictly between $-q$ and $q/2$ admits two translations, one initially approaching $-q$ and one initially moving towards the turning value $q/2$. Values below $-q$ give the unique lower family $Y=-q-(3q/2)\operatorname{csch}^2[\sqrt q(\xi+\xi_0)/2]$ with $\xi_0>0$. Matching to zero instead gives $(Y')^2=-qY^2-2Y^3/3$, which forbids a nonconstant real approach to zero. These are leading <boundary layer> classifications, not exact multiplicity theorems at degenerate endpoint data.
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