= Solution
Suppose the right-end <boundary layer> has width $\ell(\epsilon)$, and let $X=(1-x)/\ell$. An order-one change of $y$ over this width gives $\epsilon y''=O(\epsilon/\ell^2)$. Near $x=1$, the leading terms $y-x^2$ are order one. The <dominant balance> is therefore
$$
\frac\epsilon{\ell^2}\sim1.
$$
Thus
$$
\boxed{\ell=\sqrt\epsilon,\qquad X=\frac{1-x}{\sqrt\epsilon}.}
$$
The <inner variable> satisfies $X\geq0$ inside the domain and resolves the <reaction boundary layer with square-root thickness>.
Back to article page
