Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 336 / 2 / a / Solution

Away from $x=1/2$, the outer solution is obtained by setting $\epsilon =0$:It already satisfies both endpoint conditions, but its derivative jumps at $x=1/2$. Introduce the inner variableand write $y=\epsilon Y(\xi )$. The inner equation isMatching to the outer cusp requires $Y\sim |\xi |$ as $|\xi |\to \infty$. The even solution isSubtracting the common part $\epsilon |\xi |$ gives the composite asymptotic expansionIts endpoint errors are exponentially small.

At $x=1/2$, the first and second derivatives from the two sides agree, but the third derivatives have opposite signs. The composite expansion is therefore $C^2$ but not $C^3$.