= Fourier response of a first-order linear differential equation
{c}
{title2=$\widehat y_n=\widehat g_n/(a+in)$}
For $y'+ay=g$ with real $a>0$ and $2\pi$-periodic forcing, comparing <Fourier coefficients> gives $(a+in)\widehat y_n=\widehat g_n$. There is no vanishing denominator, including the constant mode. With real cosine and sine coefficients $a_n,b_n$ for $g$ and $A_n,B_n$ for $y$, the equivalent equations give
$$
A_n=\frac{a a_n-nb_n}{a^2+n^2},\qquad B_n=\frac{na_n+a b_n}{a^2+n^2}.
$$
Whenever termwise <differentiation> is justified, these formulas construct a periodic <particular solution>. For a forcing series with coefficients $O(n^{-2})$, the differentiated response coefficients are also summable, which supplies such a justification by <uniform convergence>. All other solutions add the <homogeneous solution> $Ce^{-ax}$. Since this exponential cannot be periodic unless $C=0$, the periodic solution is unique. This is also a simple example of how a linear equation changes the amplitude and phase of each harmonic independently.
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