 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/v/solution

For a monomial trial

u_{p} = A x^{m} y^{n}

, the operator gives

L u_{p} = 2 A m x^{m - 1} y^{n + 1} - A n x^{m + 1} y^{n - 1} .

Choosing

m = n = 2

matches the two powers in the forcing, and

A = 1/2

u_{p} = x^{2} y^{2} /2

u (x, y) = \frac{1}{2} x^{2} y^{2} + F (x^{2} + 2 y^{2}) .

The arbitrary function labels the values on the characteristic curves; it must have sufficient regularity for the desired classical solution.

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