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

Take the Dulac function $B=1$. Fordirect differentiation givesThe quadratic form in parentheses has matrixwhose eigenvalues are $14$ and $40$, so it is positive definite. For $0<k\le 3$ the divergence is nonpositive and vanishes only at the origin when $k=3$. The Bendixson-Dulac criterion therefore gives