13. The inner approximation.

We rescale the problem by putting


in our original differential equation


we get

The equation then transforms to


Consider the coefficients


The problem in the original equation is that the coefficient  of the highest derivative y” is small compared to the others. To avoid this problem we thus choose the main coefficient


to be of the same order as one of the other coefficients and that the other two coefficients are comparably small. We demonstrate the procedure below (remember that  is small):

Case 1):

Case 2):


Case 3):

Ve see that we only have one possibility and that is Case 1, where the main coefficient is relatively larger than the remaining coefficients. We therefore choose


Our transformed equation then becomes


If we now put  we get the equation

with the solution


The boundary condition z(0)=y(0)=0 then yields that


Our inner approximation is thus


The remaining problem is thus to determine the constant a and match the inner and outer approximations.