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Boundary layers also arise even in the absence of the first-derivative term; indeed,
equations of the form

with for can have a boundary layer at each end of the domain. (If
then the relevant part of the solution is oscillatory and boundary layers are
not present. If a (x) = 0 at an interior point, then we have a classical turning-point
problem and near this point we will require a transition layer.) The solution away
from these layers is simply given, to leading order, by y(x) = –f (x)/a(x). To see the
nature of this problem, consider the case a (x) = 1. The equation that controls the
solution in the boundary layers is then and so and
exponentially decaying solutions—ensuring bounded solutions as —arise for
A = 0 or B = 0, appropriately chosen, either on the left boundary or on the right
boundary.

E2.17 Two boundary layers
Consider the equation

for with the solution for suitable x = O(1) is written
as to leading order, where

This solution, (2.106), clearly does not satisfy the boundary conditions as nor as
The boundary layer near x = 1 is expressed in terms of
with equation (2.105) then becomes

which leads to the choice Seeking a solution then

and for bounded (i.e. matchable) solutions as we must have The
boundary condition, gives and so