The reduced outer equation is $(1+x)y_0'+y_0=0$. Imposing the right boundary condition gives
$$
\boxed{y_{\rm out}=\frac2{1+x}
+\epsilon\left[\frac2{(1+x)^3}-\frac1{2(1+x)}\right]+O(\epsilon^2).}
$$
The condition at $x=0$ requires $X=x/\epsilon$. The first two inner terms are
$$
\boxed{Y_0=2-e^{-X},
\qquad
Y_1=-2X+\frac32+\left(\frac{X^2}{2}-\frac32\right)e^{-X}.}
$$
They match $2+\epsilon(-2X+3/2)$. The <additive composite expansion> is
$$
\boxed{
y_{\rm comp}=\frac2{1+x}
+\epsilon\left[\frac2{(1+x)^3}-\frac1{2(1+x)}\right]
-e^{-x/\epsilon}
+\epsilon\left[\frac12\left(\frac x\epsilon\right)^2-\frac32\right]e^{-x/\epsilon}.}
$$

On $[-1,1]$, the coefficient $1+x$ vanishes at the left endpoint. The ordinary $O(\epsilon)$ exponential layer is replaced by a turning-point endpoint region of width $O(\sqrt\epsilon)$, where all three terms in the equation enter the leading balance.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 336 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-336/3/a/solution