= First integral of a square-root reaction layer
{title2=$(Y')^2=\tfrac23(\sqrt Y-k)^2(2\sqrt Y+k)$}
After $y=\varepsilon^2Y$ and $x=\varepsilon\xi$ or a translated version, the layer equation is $Y''-\sqrt Y+k=0$. Multiplying by $Y'$ and matching $Y=k^2$, $Y'=0$ fixes the energy constant to $k^3/3$ and gives the displayed factorization. The increasing branch below $k^2$ approaches it at positive infinity; the increasing branch above it leaves it at negative infinity and grows like $\xi^4/144$. This supplies both interfaces of the <square-root reaction problem with nested boundary layers>.
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