 Solution

Dropping the small highest-derivative term gives $y_o^{\prime }+x{y}_o=0$. The outer solution satisfying the right endpoint condition isIt fails the left endpoint condition, so a boundary layer is needed at $x=0$. If $X=x/{ϵ}^{\gamma }$, diffusion and first derivative balance when ${ϵ}^{1-2\gamma }={ϵ}^{-\gamma }$, giving $\gamma =1$.

With $y(x)=Y(X)$, the exact inner expansion after multiplication by $ϵ$ isIts leading solution is $Y=A+B{e}^{-X}$. The zero boundary value gives $B=-A$, and asymptotic matching condition as $X\to \infty$, $x\to 0$ fixes $A=e^{1/2}$. ThusThe common overlap region value is $e^{1/2}$. Add the outer and inner approximations and subtract that overlap region:The decaying inner exponential function can accommodate the left condition, whereas a right-end inner coordinate would give a mode that grows into the interval. There is no right boundary layer. The composite asymptotic expansion satisfies the left condition exactly and the right one up to the exponentially small term $e^{1/2-1/ϵ}$; the remaining approximation error is algebraic and vanishes as $ϵ\downarrow 0$.