Past exam of the differential equations 2 course of the University of Oxford / 2014 / 3 / b / iii / Solution

Suppose the right-end boundary layer has width $\ell (ϵ)$, and let $X=(1-x)/\ell$. An order-one change of $y$ over this width gives $ϵy^{\prime \prime }=O(ϵ/{\ell }^2)$. Near $x=1$, the leading terms $y-x^2$ are order one. The dominant balance is thereforeThusThe inner variable satisfies $X\ge 0$ inside the domain and resolves the reaction boundary layer with square-root thickness.