10. Inner and outer approximations.

Again consider the problem


It can be showed that the exact solution y=y(t) is


We now try to find inner and outer approximations to the problem:


1) If we let  we get the unperturbed problem


By searching for the solution that fulfills the outer boundary condition

we get an outer approximation

that well agrees with the solution for “large” (close to 1).

2) For “small” t (t close to 0, belongs to the boundary layer) a good inner approximation is 


We will in part 12 examine how to find this.