Past exam of the differential equations 2 course of the University of Oxford / 2014 / 3 / b / iv

Write $y(x)=Y(X)$ with $X=(1-x)/\sqrt{ϵ}$. The transformed equation isSince the rescaled forcing contains $\sqrt{ϵ}$, an inner expansion naturally uses powers of $\sqrt{ϵ}$. Its leading term satisfiesThe asymptotic matching condition with the leading outer solution isThe general solution is $Y_0=1+A{e}^X+B{e}^{-X}$. Matching eliminates $A{e}^X$, and the boundary condition gives $B=-1$. HenceThis tends to the boundary value zero at $x=1$ and to the matching value one on the interior side of the layer. The negative sign in the exponential follows from measuring the inner variable inward from the right endpoint.

For an optional uniform leading approximation, combine the leading inner and outer terms and subtract their common overlap value one. The additive composite expansion isIt satisfies the right boundary condition exactly; its left-end discrepancy is the exponentially small value $-e^{-1/\sqrt{ϵ}}$. The requested leading inner expansion is $Y_0$ above.