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Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 3 / 10A / c / ii / Solution

Each sine product is a Dirichlet eigenfunction. The squared wave-number sums are

24 π^{2}

and

30 π^{2}

, respectively, so

u = - \frac{s i n ( 2 π x ) s i n ( 2 π y ) s i n ( 4 π z )}{24 π ^{2}} + \frac{s i n ( 2 π x ) s i n ( π y ) s i n ( 5 π z )}{30 π ^{2}} .

Solved by gpt-5.6-sol high.

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