= Solution
Away from the thin layer, set $\epsilon=0$. The outer equation is
$$
(1+x)^3y_{\rm out}'+y_{\rm out}=0,
$$
and the boundary condition selected by this first-order problem is the right-end condition. Thus
$$
\boxed{
y_{\rm out}(x)
=\exp\left[
\frac1{2(1+x)^2}-\frac18
\right]},
$$
which satisfies $y_{\rm out}(1)=1$ but has $y_{\rm out}(0)=e^{3/8}$.
The missing left condition is supplied by an <outflow boundary layer>. Put $X=x/\epsilon$ and $y=Y(X)$. After multiplication by $\epsilon$, the equation is
$$
Y_{XX}+(1+\epsilon X)^3Y_X+\epsilon Y=0.
$$
At leading order,
$$
Y_{0,XX}+Y_{0,X}=0.
$$
The conditions $Y_0(0)=1$ and $Y_0\to y_{\rm out}(0)=e^{3/8}$ as $X\to\infty$ give
$$
\boxed{
Y_0(X)=e^{3/8}
+(1-e^{3/8})e^{-X}}.
$$
The common overlap is $e^{3/8}$. The <additive composite expansion> is therefore
$$
\boxed{
y_{\rm comp}(x)
=\exp\left[
\frac1{2(1+x)^2}-\frac18
\right]
+(1-e^{3/8})e^{-x/\epsilon}}.
$$
It satisfies both endpoint values up to exponentially small or higher-order errors and is uniformly valid to $O(1)$ on $0\leq x\leq1$.
Back to article page
