13. The inner approximation.

We rescale the problem by putting in our original differential equation If 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.