Boundary layer problem
Solution 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 $(y-1)^{-1}$