Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 17 / b / ii

The homogeneous equation is the dot product condition $(2y,-x)\cdot \nabla u=0$. A suitable vector field is thereforeWhere nonzero, it is tangent to the level curves of $u$ and perpendicular to its gradient. At the origin this vector vanishes, so no direction is defined there.