 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability/6/b/ii/solution

For the spatial mode with eigenvalue

κ_{n} = (nπ / L)^{2}

T_{n}^{'} + D (\frac{nπ}{L})^{2} T_{n} = 0.

T_{n} (t) = C_{n} exp (- \frac{D n ^{2} π ^{2}}{L ^{2}} t), C_{n} \in R .

The printed reference to “each solution

X (x)

... in part (b)” refers to (b)(i). The zero spatial mode gives

u = 0

irrespective of

T

and contributes no additional nonzero solution.^[ref]

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