Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 15 / a

Table of contents
- Solution a

Solution

The homogeneous solution is

A cos x + B sin x

. If

k \neq = 1

, the method of undetermined coefficients permits a particular integral

C cos k x

. Substitution gives

(1 - k^{2}) C = 1

and hence

y (x) = A cos x + B sin x + \frac{cos k x}{1 - k ^{2}}, k = 2, 3, \dots .

For

k = 1

, the forcing is itself a homogeneous solution, so this trial fails by resonance in a differential equation. Multiplying the sine trial by

x

gives

(\frac{d ^{2}}{d x ^{2}} + 1) (x sin x) = 2 cos x .

Thus the resonant general solution is

y (x) = A cos x + B sin x + \frac{x}{2} sin x, k = 1 .

The secular term containing

x

describes the increasing amplitude under undamped resonant forcing.

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