 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2005/calculus-linear-algebra-discrete-mathematics-logic-and-proof-and-models-of-computation/4/a/ii/solution

Δ (xg (r)) = x Δ g (r) + 2 \partial_{x} g (r) = x (g^{''} + \frac{g ^{'}}{r}) + 2 \frac{x}{r} g^{'} = x (g^{''} + \frac{3 g ^{'}}{r}) .

Thus Laplace's equation, away from the origin, reduces to

g^{''} + 3 g^{'} / r = 0

. On the line

x = 0

the displayed expression vanishes automatically; requiring the equation on an open region gives the same radial equation by continuity from

x \neq = 0

. Multiplying by

r^{3}

gives

(r^{3} g^{'})^{'} = 0

, so

g^{'} = C r^{- 3}

and

g = A + B r^{- 2}

. Therefore

u (x, y) = A x + B \frac{x}{x ^{2} + y ^{2}}, r > 0.

If extension through the origin is required,

B = 0

: along the horizontal axis the second term is

B / x

. The remaining harmonic function is

A x

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