 Solution

For the outer expansion, the first-order reduced ordinary differential equation gives $w_0=e^{-x^2}$ after imposing the right boundary condition. Its first correction satisfiesAn integrating factor givesThe left boundary is at $x=\epsilon$, where the highest-derivative coefficient is already ${\epsilon }^2$. Balancing that coefficient against $w^{\prime }$ gives a layer of width ${\epsilon }^2$, not $\epsilon$. This is a boundary layer around a moving endpoint:At leading order $W_0^{\prime \prime }+W_0^{\prime }=0$, and matching to the outer limit one while setting $W_0(0)=0$ gives $W_0=1-e^{-z}$. The first correction satisfiesDirect differentiation of $-z(z+2)e^{-z}/2$ verifies the forcing. It vanishes both at zero and at infinity, so no homogeneous term is needed. The requested inner expansion isIts order-$\epsilon$ overlap constant is zero, agreeing with the outer expansion at the moving endpoint. The overlap can be chosen with $1\ll z\ll {\epsilon }^{-1}$.

An independent scaling check drops the reaction term only in the layer and integrates $\epsilon x{w}^{\prime \prime }+w^{\prime }=0$. Its decaying fast solution is proportional to $(x/\epsilon {)}^{1-1/\epsilon }$. Sinceit gives exactly the same first correction. A useful additive composite expansion, with the left boundary value made exact, isThe exponentially small remainder at the right endpoint is beyond all retained algebraic orders.