Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 336 / 2 / a

Use the fast time $t$ and slow time $T={ϵ}^{2}t$, and writeThere is no $O(ϵ)$ resonant forcing. Solvinggives a convenient particular solutionAt $O({ϵ}^2)$, the coefficient of $e^{it}$ in the forcing must vanish by the solvability condition in the method of multiple scales. This givesorThe two slow exponents satisfyThe second instability tongue of a weak Mathieu oscillator has real $s$ when $-1/12<f<5/12$. At either endpoint the repeated zero exponent permits a linearly growing slow solution, so boundedness for every initial condition requires the strict stable ranges