Geodesics on R^2

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Hello,

Suppose that [tex]R^2[/tex] is provided with the following metric

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

The geodesic equation gives
[tex]
x'' - \cosh(x)\sinh(x) (y')^2 = 0
[/tex]
[tex]
y'' + 2 \tanh(x) x' y' = 0
[/tex]

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

Thank you
 

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