Consider the boundary value problem $$\varepsilon \frac{d^2y}{dx^2}+(1+x)\frac{dy}{dx}+y=0$$ subject to $y(0)=0$, $y(1)=1$, for $0 \leq x \leq 1$.
Use the method of matched asymptotic expansions to construct two-term inner and outer expansions to the problem, which should then be matched using Van Dyke's matching principle.
What i did so far is:
$y \sim y_0 +y_1 \varepsilon + O(\varepsilon ^2) $
$$\varepsilon \frac{d^2y_0}{dx^2}+\varepsilon ^2y_1+(1+x)\bigg(\frac{dy_0}{dx}+ \varepsilon \frac{dy_1}{dx} \bigg)+y_0 +\varepsilon y_1=0$$
At $O(1)$: $$ (1+x)\frac{dy_0}{dx}+y_0=0 $$ giving $y_0=2(1+x)^{-1}$
At $O(\varepsilon )$: $$\frac{d^2y_0}{dx^2}+(1+x) \frac{dy_1}{dx} +y_1=0 $$ giving $y_1=2(1+x)^{-3}-\frac12(1+x)^{-1}$
What do I do next?
Inner solution: $$Y \sim -B_0 +B_0 e^{-X} + \varepsilon \bigg( -\frac12 B_0 X^2 e^{-X} - ke^{-X} +B_0 X +k \bigg) + O(\varepsilon ^2)$$
Outer solution: $$y \sim 2(1+x)^{-1} +2 \varepsilon (1+x)^{-3} - \frac{\varepsilon}2(1+x)^{-1} + O(\varepsilon ^2)$$
Van Dyke:
Write inner in terms of outer: $$y \sim B_0 (x-1) + \varepsilon k + O(\varepsilon ^2)$$
Write outer in terms of inner (after expanding terms by binomial): $$ Y \sim 2+ \varepsilon (3 /2 -2X) +O(\varepsilon ^2)$$
Matching them gives $B_0 =-2$ and $k=3/2$