15. A final example.

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




We compare the leading coefficient


with the other ( is supposed to be small):

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