Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 5 / i

Table of contents
- Solution i

Solution

The Fourier transform of

u_{xx}

is

- k^{2} u

: two integrations by parts have no boundary terms under the stated decay assumption. Time differentiation commutes with the spatial transform. Therefore

u_{tt} + 2 γ u_{t} + (γ^{2} + c^{2} k^{2}) u = 0.

For every fixed spatial wavenumber

k

, this is a constant-coefficient ordinary differential equation in time.

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