Geodesics on R^2: Exact Formula?

[Statis]

Hello,

Suppose that R^2 is provided with the following metric

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

The geodesic equation gives
<br /> x&#039;&#039; - \cosh(x)\sinh(x) (y&#039;)^2 = 0 <br />
<br /> y&#039;&#039; + 2 \tanh(x) x&#039; y&#039; = 0<br />

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