Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 32E / b / i

Table of contents
- Solution i

Solution

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),

whose eigenvalues are

14

and

40

, so it is positive definite. For

0 < k \leq 3

the divergence is nonpositive and vanishes only at the origin when

k = 3

. The Bendixson-Dulac criterion therefore gives

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

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