 Solution

ID: 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/solution

The homogeneous equation is the dot product condition

(2 y, - x) \cdot \nabla u = 0

. A suitable vector field is therefore

v (x, y) = (2 y, - x) .

Where 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.

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