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 Incoming links: Fourier response of a first-order linear differential equation

Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 4 / b / iv / Solution 2026-10-31

Seek a periodic particular solution with Fourier coefficients

A_{n}, B_{n}

. Comparing constant terms gives

A_{0} = a_{0} /2

. Comparing the cosine and sine terms for

n \geq 1

gives

2 A_{n} + n B_{n} = a_{n}, - n A_{n} + 2 B_{n} = 0,

so

A_{n} = 2 a_{n} / (n^{2} + 4)

and

B_{n} = n a_{n} / (n^{2} + 4)

. Insert the coefficients from part (a) and add the homogeneous solution:

y (x) = C e^{- 2 x} + \frac{1}{π} - \frac{2}{π} m = 1 \sum \infty \frac{cos ( 2 m x ) + m sin ( 2 m x )}{( 4 m ^{2} - 1 ) ( m ^{2} + 1 )} .

The differentiated series converges absolutely and uniformly, because its largest coefficient terms are

O (m^{- 2})

. This justifies the term-by-term first derivative used in substituting into the differential equation. Any two solutions differ by

C e^{- 2 x}

, proving that the displayed family is the general solution. The computation is an instance of Fourier response of a first-order linear differential equation.

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