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 Incoming links: Fourier solution of a first-order equation with periodic forcing

The constant coefficient is

a_{0} = 0

. For every

n \geq 1

the two mode equations are

a_{n} + 3 n b_{n} = 0, b_{n} - 3 n a_{n} = 4 δ_{n 1},

where

δ_{n 1}

a_{n} = - 3 n b_{n}

, so the second becomes

(1 + 3 n^{2}) b_{n} = 4 δ_{n 1}

. Hence

a_{0} = 0, a_{1} = - 3, b_{1} = 1, a_{n} = b_{n} = 0 (n \geq 2) .