In the overlapping region we let

and introduce the *intermediate variable*

This is a time scale that is “between” de inner

and outer

time scales. To be able to *match* the approximations we require that the outer and inner approximations (written with respect to the intermediate variable) must agree in the limit

that is

for a fix (positive) value of . in our case this means that

VWe realize that *a=e* and that our inner approximation therefore must be

Finally, we want to find a solution that is valid in all the interval *[0,1]*. We therefore construct *y*_{u} from the inner and outer approximations *minus* their common limit *e* (since it otherwise would have been counted twice) in the overlapping region

When *t* is in the outer region the second term is small and *y*_{u} is thus approximately

which is exactly the outer approximation. When *t* is in the boundary layer the first term is close to *e* and *y*_{u} is thus approximately

which is exactly the inner approximation. In the overlapping region both the inner and outer approximations are approximately equal to *e*, which makes the sum of *y*_{i} and *y*_{o} close to *2e *there, that is, twice as much as it should be. That is why we have to subtract the common limit from the sum. If we insert *y*_{u} in the original differental equation we see that

dthat is, *y*_{u} fulfills the equation exactly on the interval *(0,1)*. If we investigate the boundary conditions we see that

The left condition is exactly fulfilled and the right is fulfilled up to

´for any *n>0* , since

We thus see that *y*_{u} is a unitly good approximation on *[0,1]*.