Geodesics of conformal metrics in complex domains Let $U$ be  a non-empty domain in the complex plane $\mathbb C$. 
Question: what is the differential equation of the geodesics of the metric 
$$m=\varphi(x,y) (dx^2+dy^2)$$ where $\varphi$ is a positive function on $U$ and where 
$x,y$ are the usual euclidian coordinates on $\mathbb C\simeq \mathbb R^2$
Certainly, an answer can be found in many classical textbooks. 
But I'm interested in the (simpler) case when $\varphi=\lvert f(z)\lvert^2$ where 
$f$ is a holomorphic function of $z=x+iy$.  And I didn't find in the classical literature a simple differential equation characterizing geodesics for metric of the form 
$$m= \lvert f(z) dz\lvert^2.$$
Does anyone know the answer or a good reference? 
Many thanks in advance.

You should be worried about the zeroes of $f$; the geodesic equation degenerates at the points where the metric vanishes. 
At the points where $f\ne 0$ the local structure of geodesics is indeed simple. Let $F$ be an antiderivative of $f$, that is $F'=f$. The metric $|F'(z)|^2\,|dz|^2$ is exactly the pullback of the   Euclidean metric under the map $F$. Therefore, geodesics are preimages of straight lines under $F$; note that $F$ is locally a diffeomorphism because $F'\ne 0$. Stated another way, geodesics are curves of the form $\mathrm{Re}\,(e^{i\theta}F)=c$ for some $\theta,c\in\mathbb R$. If you want a differential equation for parametrized geodesics $t\mapsto z(t)$, it is $\dfrac{d}{dt}\mathrm{Re}\,(e^{i\theta}F(z(t)))=0$.
Example: two orthogonal families of geodesics for  $f(z)=z^2$, that is, the metric $|z|^4|dz|^2$
