Solution
= Solution
Insert the <exponential function> trial $y=e^{rx}$ into the <homogeneous linear differential equation>. Its <characteristic equation> is $r^2+\sqrt3r+3=0$, with <polynomial roots>
$$
r=-\frac{\sqrt3}{2}\pm\frac32i.
$$
Taking real combinations of the two complex exponential solutions produces the <general solution>
$$
\boxed{y_c(x)=e^{-\sqrt3x/2}\left(A\cos\frac{3x}{2}+B\sin\frac{3x}{2}\right).}
$$