Solution
= Solution
Put $z=x-ct$ and $h(x,t)=f(z)$. The conservation law becomes
$$
-cf'
+\frac d{dz}\left[
\frac{\rho gf^3}{3\mu}
(\sin\alpha-\cos\alpha\,f')
\right]=0.
$$
One integration gives the required first-order autonomous equation
$$
\boxed{
\frac{\rho gf^3}{3\mu}
(\sin\alpha-\cos\alpha\,f')
-cf=C},
$$
where $C$ is a constant fixed by the far-field film heights or other boundary data.