 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2002/paper-1/12/solution

For

k > 0

h (x) = A cosh k x + B sinh k x, g (x) = C cos k x + D sin k x .

Their products

h (x) g (y)

and

g (x) h (y)

solve Laplace's equation, since the two second derivatives contribute opposite

k^{2}

terms. For

k = 0

, both factors are linear functions. This is separation of variables with positive, negative and zero separation constants.

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