Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 2 / 8A / a / Solution

The system matrix has eigenvalues $2$ and $3$, with eigenvectors in the directions $y=-x$ and $y=-x/2$, respectively. Both eigenvalues are positive, so the origin is an unstable node. The two eigendirections are straight trajectories directed away from the origin. Every other trajectory also moves outward, tangent near the origin in backward time to the faster eigendirection and asymptotic in forward time to the eigenvalue-$3$ direction.