Let the ordered pair $(x,y)\in\mathbb{R}^2$ be the unique point in the hyperbolic plane arrived at by starting at [an arbitrary point called the origin] and going [east, an arbitrarily chosen direction] distance $x$ followed by going [north, an arbitrarily chosen direction perpendicular to east] distance $y$. $x\in\mathbb{R}$ and $y\in\mathbb{R}$ are hyperbolic distances. Every point in the hyperbolic plane can be reached in this method. A little work showed that the metric of the plane is $$(ds)^2=(\cosh(y)*dx)^2+(dy)^2$$ Let $y=g(x)$ be a geodesic. Then the Euler-Lagrange formula leads to the equation $$\tanh(g(x))(\cosh(g(x))^2+2g'(x)^2)=g''(x)$$ Also, $y=\pm g(\pm x-c)$ is a geodesic for $c\in\mathbb{R}$, as can be seen by the symmetry of the metric.
All geodesics can be divided into 4 types of geodesics:
The trivial case $y=0$.
Those that do not cross the x-axis, looking similar to $c_0\sec(c_2(x-c_1))$. $$g_0(0)\in\mathbb{R}^+,g_0'(0)=0,g_0(x)=g_0(-x),y=\pm g_0(x-c_0),c_0\in\mathbb{R}$$
Those that cross the x-axis, looking similar to $c_0\tan(c_2(x-c_1))$. $$g_1(0)=0,g_1'(0)\in\mathbb{R}^+,g_1(x)=-g_1(-x),y=\pm g_1(x-c_1),c_1\in\mathbb{R}$$
Those that touch the x-axis once infinitely far from the origin. $$\lim_{x\to -\infty}g_2(x)=0,\lim_{h\to\infty}g_2(-1/h)=\infty,y=\pm g_2(\pm x-c_2),c_2\in\mathbb{R}$$
Please help find the formulae for $g_k(x)$ in these 3 non-trivial cases.
More details on the derivation of the metric and of the Euler-Lagrange result are here.