= Solution
The constant <coefficient> is $a_0=0$. For every $n\geq1$ the two mode equations are
$$
a_n+\sqrt3\,n b_n=0,\qquad b_n-\sqrt3\,n a_n=4\delta_{n1},
$$
where $\delta_{n1}$ is the <Kronecker delta>. The first equation gives $a_n=-\sqrt3\,n b_n$, so the second becomes $(1+3n^2)b_n=4\delta_{n1}$. Hence
$$
\boxed{a_0=0,\qquad a_1=-\sqrt3,\quad b_1=1,\qquad a_n=b_n=0\ (n\geq2).}
$$
No other modes survive. This is the single-frequency case of the <Fourier solution of a first-order equation with periodic forcing>.
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