The metric is $ds^2=\frac{dx^2+dy^2}{y^2}$. I have used the Euler-Lagrange equations to find the geodesics, and my equations are $\dot{x}=Ay^2$, $\ddot{y}+\frac{\dot{x}^2-\dot{y}^2}{y}=0$. I cannot seem to find the first integral for the second equation. I read online somewhere it is $\dot{y}=y\sqrt{1-Ay^2}$, but I can't seem to derive it. The only trick I currently know for doing these type of things is to multiple by $\dot{y}$ and then integrate, but that doesn't work here. Can anyone offer some guidance?
I tried it a slightly different way, but it doesn't seem to work for some reason: Instead of parametrizing, x=x(t), y=y(t) and minimizing, I just minimized $\frac{1+y'(x)^2}{y^2}$. Using the Euler-Lagrange equations, I get $y''y-y'^2+1=0$, and $y(x)=sinh(x)$ is a solution to this...but the geodesics are suppose to be half circles, and this doesn't give me a half circle..I am quite confused.