= Fourier solution of a first-order equation with periodic forcing
{c}
{title2=$\alpha y'+\beta y=f$}
Let $\alpha,\beta$ be nonzero real constants and let $f$ be a <smooth> <periodic function> of period $2\pi$, with real <Fourier coefficients> $c_n,d_n$. Write the unknown <periodic solution> with <Fourier coefficients> $a_n,b_n$. <Fourier orthogonality> applied to the equation gives
$$
\beta a_0=c_0,\qquad
\beta a_n+\alpha n b_n=c_n,\qquad
\beta b_n-\alpha n a_n=d_n.
$$
Solving these two-by-two <linear systems> gives
$$
a_n=\frac{\beta c_n-\alpha n d_n}{\beta^2+\alpha^2n^2},\qquad
b_n=\frac{\alpha n c_n+\beta d_n}{\beta^2+\alpha^2n^2}\quad(n\geq1).
$$
The smooth forcing has rapidly decreasing <Fourier coefficients>, so the resulting series and its <derivatives> converge and satisfy the equation. Uniqueness follows because the difference of two solutions is $Ce^{-\beta x/\alpha}$, which is periodic only when $C=0$. A single sinusoidal forcing therefore gives a single sinusoidal response with changed amplitude and phase.
Back to article page
