 Solution

For $\sigma =1$, the formal outer family isEvery nonzero member diverges as $x\to 1^{-}$, so bounded matching selects the outer solution $z_0=0$. Boundary layers are now required at both endpoints.

The $x=1$ layer still has width $O(\epsilon )$ and leading profileNear $x=0$, diffusion and the logarithmically vanishing convection balance on the thinner scaleAt leading order the reaction term is smaller there. Using the supplied first integral with $\mu ={\epsilon }^2$, defineThe left layer that equals $1$ at the endpoint and matches zero isConsequently a leading composite description isIts sketch has value $1$ at each endpoint, drops sharply to an almost-zero outer plateau just to the right of $x=0$, and rises through an $O(\epsilon )$ layer just before $x=1$.