Example: Consider the differential equation
We get the outer approximation by putting 
and solving the equation with the outer boundary condition:
that is,
We now look for the inner approximation. Put
The equation is then transformed to
where
We compare the leading coefficient
with the other (
is supposed to be small):
| Case1): |  |
 |
 |
| Case 2): |  |
 |
 |
We see that in Case 2, the remaining coefficient is much smaller than the other two. We therefore choose
and make the change of variables
Our equation then becomes
We get the inner approximation when we put and solve the equation:
which in the original variables is
The condition y(0)=1 then gives
Let us match these approximations. Introduce the intermediate variable
The matching condition
then implies that
that is,
Our inner approximation finally becomes
We get a unit approximation yu by adding the inner and outer approximations and subtracting their common limit in the overlapping region
