 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2008/paper-2/14/b/ii/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

For

k \neq = 0

, the chain rule gives

g_{t} = - f_{t} g / (2 k)

and

g_{xx} = (- f_{xx} / (2 k) + f_{x}^{2} / (4 k^{2})) g

. Consequently

g_{t} - k g_{xx} = - \frac{g}{2 k} (f_{t} + \frac{1}{2} f_{x}^{2} - k f_{xx}) = 0.

Thus

g_{t} = k g_{xx} .

This is the potential form of the Cole-Hopf transformation, with

y = - 2 k g_{x} / g

. For

k > 0

it is a forward diffusion equation; at

k = 0

the exponential transformation is undefined, while negative

k

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