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?