Perturbative Methods for Modelling (M2)

Example 1, p. 419

We consider the differential equation $\fs2 \epsilon \, y'' +(1+\epsilon)\,y'+y=0$ with the boundary conditions $\fs2 y(0)=0$ and $\fs2 y(1)=1$. Find the exact solution.

For $\fs2 \epsilon\to 0$, show that the outer solution satisfies $\fs2 y'_{out}+y_{out}=0$ with $\fs2 y_{out}(0)=0$ and the inner solution is such that $\fs2 Y''_{in} + Y_{in} =0$ with $\fs2 Y_{in}(0)=0$ and $\fs2 Y_{in}(+\infty)=e$ with $\fs2 y_{in}(x) = Y_{in}(X)$  and $\fs2 x=\epsilon\, X$. Draw the matching  between the inner and outer solutions. Write a uniform approximation  under the form $\fs2 y_{unif} = y_{in} + y_{out} + y_{match}$.