 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-3/41e/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

r^{n + 1} = r^{n - 1} - 2 q r^{n}, p (r) = r^{2} + 2 q r - 1.

The two amplification roots are

r_{\pm} = - q \pm 1 + q^{2} .

For every nonzero mode

q > 0

, one has

∣ r_{-} ∣ = q + 1 + q^{2} > 1.

Thus every

μ > 0

admits a growing Fourier mode, and

the leapfrog scheme is unstable for every μ > 0.

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