(i) Give a brief description of the method of matched asymptotic expansions, as applied to a differential equation of the type

$$\epsilon y^{\prime \prime}+K y^{\prime}+f(y)=0, \quad 0<x<1,$$

where $0<\epsilon \ll 1, K$ is a non-zero constant, $f$ is a suitable smooth function and the boundary values $y(0), y(1)$ are specified. An outline of Van Dyke's asymptotic matching principle should be included.

(ii) Consider the boundary-value problem

$$\epsilon y^{\prime \prime}+y^{\prime}-(2 x+1) y=0, \quad y(0)=0, \quad y(1)=e^{2}$$

with $0<\epsilon \ll 1$. Find the integrating factor for the leading-order outer problem. Hence obtain the first two terms in the outer expansion.

Rewrite the problem using an appropriate stretched inner variable. Hence obtain the first two terms of the inner exansion.

Use van Dyke's matching principle to determine all the constants. Hence show that $y^{\prime}(0)=\epsilon^{-1}+\frac{25}{3}+O(\epsilon) .$