Nonlinear Systems & Linearization · Challenge problem

The system $x' = xy,\; y' = -y - x^2$ has an equilibrium at the origin whose linearization has eigenvalues $0$ and $-1$, so linearization cannot decide its stability. In fact every solution that starts close enough to the origin with $x(0) > 0$ tends to the origin, but slowly. For such a solution, find $\displaystyle\lim_{t\to\infty}\sqrt{t}\,x(t)$.