Boundary layer problem

Question

This question is taken from Bender & Orszag "perturbation methods"

$y' = (1 + X^{-2}/100)y^2 - 2y + 1$ ,$y(1)=1$

first we can see that if we set $\epsilon=100x^{2}$ we can translate the above to the boundary layer problem

$\epsilon y' = (\epsilon+1)y^{2}-\epsilon y+\epsilon$

my question is what is the outer and the inner solution on [0,1]?

Thanks

Answer 1

This is not a boundary layer perturbation problem. The proposed $ϵ=100x^2$ is not a small term around $x=1$.

One reduced equation is $$ y'=y^2-2y+1=(y-1)^2 $$ with solution $$ y\equiv 1\text{ or }y=1-\frac{1}{x-c}. $$ Now one can add perturbation terms either to $y$ or to $(y-1)^{-1}$