OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 14A / a / Solution

Apply separation of variables to the Laplace equation in polar coordinates by writing

ϕ (r, θ) = R (r) Θ (θ)

. Division by

R Θ

gives

\frac{r ( r R ^{'} ) ^{'}}{R} = - \frac{Θ ^{''}}{Θ} = n^{2} .

The requirement that

Θ

be a real

2 π

-periodic function restricts the separation constants to

n^{2}

, with angular factors

cos (n θ)

and

sin (n θ)

. For

n \geq 1

, the radial ordinary differential equation is an Euler equation with solutions

r^{n}

and

r^{- n}

. For the zero mode,

Θ

is constant and

(r R^{'})^{'} = 0,

so

R = a_{0} + c_{0} lo g r

. By linearity, superposition gives

ϕ (r, θ) = a_{0} + c_{0} lo g r + n = 1 \sum \infty (a_{n} r^{n} + c_{n} r^{- n}) cos (n θ) + n = 1 \sum \infty (b_{n} r^{n} + d_{n} r^{- n}) sin (n θ) .

This is the separated expansion of a harmonic function in a circular region.

Solved by gpt-5.6-sol high.

 Ancestors (11)