# How to know which boundary condition to use

With asymptotic methods for ODEs where you have like an inner, outer region and you are given two boundary condition, how do you know which condition to use when constructing the inner/outer solution?

For example, $$ε \frac{d^2y}{ dx^2} + (1 + x) \frac{dy }{dx} + y = 0$$ subject to $y(0) = 0$, $y(1) = 1$, for $0 \le x \le 1$, $ε ≪ 1$.

The boundary layer is at $x=0$.

Use the method of matched asymptotic expansions to construct two-term inner and outer expansions to the problem, which should then be matched using Van Dyke’s matching principle.

How do we know which condition to use for each region?

You are told that the boundary layer is at $x=0$.
The inner solution is the solution in the boundary layer, so for this problem you use the boundary condition at $x=0$ for the inner solution since that's where the boundary layer is.
The outer solution is the solution away from the boundary layer, so for this problem you use the boundary condition at $x=1$ for the outer solution since that's outside of the boundary layer.