 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-2/32e/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Take the Dulac function

B = 1

. For

Q = 5 x^{2} - 6 x y + 5 y^{2},

direct differentiation gives

\frac{\partial x ˙}{\partial x} + \frac{\partial y ˙}{\partial y} = 2 (k - 3) - 2 (27 x^{2} - 26 x y + 27 y^{2}) .

The quadratic form in parentheses has matrix

(27 - 13 - 13 27),

14

and

40

0 < k \leq 3

the divergence is nonpositive and vanishes only at the origin when

k = 3

there is no periodic orbit for 0 < k \leq 3 .

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