Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 336 / 3 / a

Use the outer expansionsThe reduced leading equation isFor the symmetric fundamental mode, $y_0=a\cos ({\omega }_{0}x)$. The reduced second-order problem can impose the displacement conditions $y_0(\pm 1)=0$ but not both clamped-slope conditions. The smallest positive root is

Near the right endpoint introduce the stretched coordinateand near the left endpoint use $X_{-}=(1+x)/ϵ$. The leading outer solution is $O(ϵ)$ in either inner region, so put $y=ϵY_0+\cdots$. The dominant inner equation isAt either endpoint, clamping gives $Y_0(0)=Y_0^{\prime }(0)=0$, bounded matching excludes the growing exponential, and the outer behavior requiresThus the two leading inner solutions have the same form:They show explicitly that each clamped endpoint has a boundary layer of width $O(ϵ)$.

At order $ϵ$, the outer equation isThe symmetric solution iswhich is the stated resonant particular solution plus a homogeneous normalization term.

As $x=1-ϵX_{+}$ approaches the right endpoint from the outer region,The large-$X_{+}$ inner expansion isMatching the constant terms givesThe left endpoint gives the same condition by symmetry. Therefore