Geodesics on R^2: Exact Formula?

[Statis]

Hello,

Suppose that $$R^2$$ is provided with the following metric

$$ds^2 = dx^2 + (\cosh(x))^2 dy^2$$
Can we find a general exact formula $$\alpha(t)$$ for the geodesics (starting at an arbitrary point) ?

The geodesic equation gives
$$x'' - \cosh(x)\sinh(x) (y')^2 = 0$$
$$y'' + 2 \tanh(x) x' y' = 0$$

I guess that since this model is simply a reparametrization of the Hyperbolic space on R^2 the geodesics should be known ?

Thank you

 

[fresh_42]

I guess that since this model is simply a reparametrization of the Hyperbolic space on R^2 the geodesics should be known ?

Yes. http://math.uchicago.edu/~may/REU2014/REUPapers/Adkisson.pdf