Nonlinear Systems & Linearization · Challenge problem

The system $x' = x(1 - x),\; y' = x - y$ has a saddle at the origin and a stable node at $(1, 1)$. The branch of the origin's unstable manifold that enters the region $x > 0$ runs all the way to $(1, 1)$. At what height $y$ does this heteroclinic orbit cross the line $x = \tfrac12$? Give an exact answer.