Curvature and geodesic of $M = \mathbb{R}\times \mathbb{R}_{>0}$ with metric $ds^2 = dx^2 + ydy^2$

Hint for part b

(IMPORTANT: in the following context, direct superscripts on symbols do not imply raising to powers.)

The geodesics equations are $$ \frac{\partial^2 x^i}{\partial s^2}+\Gamma_{\alpha\beta}^i\frac{\partial x^\alpha}{\partial s}\frac{\partial x^\beta}{\partial s}=0, $$ where $i,\alpha,\beta \in \{1,2\}$. In the case of your question, the geodesics are found among $$ {\frac{\partial^2 x}{\partial s^2}=0, \\ \frac{\partial^2 y}{\partial s^2}+\frac{1}{2y}(\frac{\partial y}{\partial s})^2=0. } $$

Additional Remark

I couldn't fully conceive what $\Gamma_{vv}^v = \frac{1}{2v}$ means. Unless I'm missing something, the expression $\Gamma_{vv}^v = \frac{1}{2v}$ means $$ { \Gamma_{11}^1 = \frac{1}{2}, \\\Gamma_{22}^2 = \frac{1}{4}, } $$ which is indeed incorrect. Actually, you should write the only non-zero Christoffel symbol as $$ \Gamma_{22}^2 = \frac{1}{2x^2}=\frac{1}{2y}. $$