 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/a/solution

w_{xx} = \frac{180 x ^{2} y ^{2} - 30 x ^{4} + 30 y ^{4}}{360}, w_{yy} = \frac{180 x ^{2} y ^{2} + 30 x ^{4} - 30 y ^{4}}{360} .

Adding cancels the pure fourth powers and leaves

w_{xx} + w_{yy} = x^{2} y^{2} .

Thus direct substitution verifies the stated Poisson equation.

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