Solution
= Solution
Substitute the monochromatic wave
$$
\phi=e^{i(kx-\omega t)}
$$
into the differential equation. Since $(ik)^9=ik^9$, cancellation of the common exponential gives
$$
-i\omega+iUk+\frac{i}{9}k^9=0.
$$
Thus the <dispersion relation> is
$$
\boxed{\omega(k)=Uk+\frac{k^9}{9}.}
$$