I'm having trouble to understand a very simple fact of the book of DoCarmo "Riemannian Geometry". In the page 73 he calculates the geodesics of the hyperbolic plane:
$$ \mathbb{R}^+\_2 = \{ (x,y) \in \mathbb{R}^² :y>0 \} \qquad g\_{ij}= y^{-2} \delta\_{ij} $$
To do so he considers the curve $\gamma(t)=(0,t)$ which is a geodesic. But when I consider the equations of the geodesics I don't get this conclusion. 

The Cristoffel symbols for the hyperbolic plane with this metric are:

$$\Gamma^x\_{xx}=\Gamma^y\_{xy}=\Gamma^x\_{yy}=0 \quad \Gamma^y\_{xx}=\frac{1}{y} \quad \Gamma^x\_{xy} = \Gamma^y\_{yy} = -\frac{1}{y}$$

And then the equations of the geodesics:

$$ \frac{d^2x}{dt^2} - \frac{1}{y}\frac{dx}{dt} \frac{dy}{dt} = 0 $$
$$ \frac{d^2y}{dt^2} + \frac{1}{y} \left( \frac{dx}{dt} \right)^2 - \frac{1}{y} \left( \frac{dy}{dt} \right)^2 = 0$$

Then when I consider the given curve it satisfies the first equation but not the second. What I am missing here?