Logarithmic matching in a small-exponent boundary layer ($y(0)=1+ϵ\log ϵ+O({ϵ}^2{\log }^2ϵ)$)

In a singular first-order equation with coefficient $x+ϵ$, an outer correction proportional to $ϵ\log x$ cannot be evaluated at the endpoint. The inner scale $x=ϵX$ changes it to $ϵ\log ϵ+ϵ\log (1+X)$. Matching retains the logarithmic constant, which is larger than a plain order-epsilon correction. For $(x+ϵ)y^{\prime }=ϵy$, $y(1)=1$, the exact solution $[(x+ϵ)/(1+ϵ){]}^{ϵ}$ verifies the displayed endpoint asymptotic and its remainder.