Fourier response of a first-order linear differential equation (${\hat{y}}_n={\hat{g}}_n/(a+in)$)

For $y^{\prime }+ay=g$ with real $a>0$ and $2\pi$-periodic forcing, comparing Fourier coefficients gives $(a+in){\hat{y}}_n={\hat{g}}_n$. 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 giveWhenever 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 $C{e}^{-ax}$. 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.