Find uniform approximation up to order $O(\epsilon)$: $$ \begin{cases} \epsilon y''+\epsilon y' - y^2=-1-x^2 \\ y(0)=2 \\ y(1)=2 \end{cases} $$ At $\epsilon=0$ solutions $\pm \sqrt{1+x^2}$ don't satisfy either of the B.C. Does this mean there are two boundary layers? E.g. to find the scaling of the boundary layer near $x=0$ let $y(x)=u(\epsilon^{\alpha}x)$ we get $$\epsilon^{2\alpha +1} u''+\epsilon^{\alpha +1} u' -u^2 = -1 - \epsilon^{-2\alpha} \bar{x}^2$$
where $\bar{x}=\epsilon^{\alpha}x$ is the new variable. Seems the only balance is when $\alpha=-\frac{1}{2}$ then we get (for the leading order term) $$ \begin{cases} u''-u^2=-1 \\ u(0)=2 \end{cases} $$ Don't know how to further analyze this problem. (Another possible balance is $\alpha=-1$ but this doesn't match the outer solution)