I have the metric $$ds^2=\frac{1}{y^2}(dx^2+dy^2)$$ and I want to find the geodesics. I know this is the metric of the hyperbolic plane which has geodesics $$\dot{x}^2+\dot{y}^2=y^2$$ and $$\dot{x}=ky^2$$, for some constant $k$ and parameterized with $s$,
but when I use the formula for the geodesics, namely $$(g_{kc} \dot{x}^k)^{\cdot}=\frac{1}{2}g_{ab,c} \dot{x}^a \dot{x}^b$$ where $g_{ab,c}=\frac{ \partial g_{ab}}{\partial x^c}$, $x^1=x$, $x^2=y$ and the dot is differentiation with respect to $s$.
For $c=1$ I get $$\dot{x}=ky^2$$ but for $c=2$ I get something completely different, $$\ddot{y}y=-\dot{x}^2+\dot{y}^2$$ if I got the calculations right. So I get different geodesic equations than the ones expected. Now I understand how $$\dot{x}^2+\dot{y}^2=y^2$$ is derived, I can get this by simply dividing the metric with $ds^2$ but my question is, why do I get different geodesic equations using the formula?