If $\gamma(s) = \mathbf{x}(\gamma^1(s),\gamma^2(s))$ where $\mathbf{x}$ is a coordinate patch, then what is the differential equations that $\gamma^k$ ($k = 1,2$) must satisfy if $\gamma$ is a geodesic?
Note: A unit speed curve $\gamma(s)$ in $M$ is a geodesic iff $[n,T,T'] = 0.$
I'll get you started.
I'm going to assume that $\gamma$ is a unit-speed curve. I'm also going to use the notation $\dot{f}$ to mean $\frac{df}{ds}$.
Although I could write $\mathbf{x}(u,v) = (x^1(u,v), x^2(u,v), x^3(u,v))$, I'm going to avoid doing that because it would make our calculations even messier.
By the chain rule:
$$T(s) = \gamma'(s) = \mathbf{x}_u\dot{\gamma}^1 + \mathbf{x}_v\dot{\gamma}^2$$
Again by the chain rule (and product rule):
$$\begin{align*} T'(s) & = \frac{d}{ds}\left[ \mathbf{x}_u\dot{\gamma}^1 + \mathbf{x}_v\dot{\gamma}^2 \right] \\ & = \left[\mathbf{x}_u\ddot{\gamma}^1 + \mathbf{x}_{uu}(\dot{\gamma}^1)^2 + \mathbf{x}_{uv}\dot{\gamma}^1\dot{\gamma}^2\right] + \left[\mathbf{x}_v\ddot{\gamma}^2 + \mathbf{x}_{vv}(\dot{\gamma}^2)^2 + \mathbf{x}_{uv}\dot{\gamma}^1\dot{\gamma}^2\right] \\ & = \mathbf{x}_u\ddot{\gamma}^1 + \mathbf{x}_{uu}(\dot{\gamma}^1)^2 + 2\mathbf{x}_{uv}\dot{\gamma}^1\dot{\gamma}^2 + \mathbf{x}_v\ddot{\gamma}^2 + \mathbf{x}_{vv}(\dot{\gamma}^2)^2. \\ \end{align*}$$
By multivariable calculus, $$n = \frac{\mathbf{x}_u \times \mathbf{x}_v}{|\mathbf{x}_u \times \mathbf{x}_v|}.$$
To get the differential equation we want, we're going to use the definition of "geodesic" -- namely $[n, T, T'] = 0$. That is, $$(n \times T) \cdot T' = 0.$$ This means that we have to compute $n \times T$, then take its dot product with $T'$, and finally set all of that equal to zero.
I leave the rest to you.
Possibly useful: When computing the cross product, it might be helpful to use the formula $$\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a}\cdot \mathbf{b})\mathbf{c},$$ and to remember that $g_{11} = \mathbf{x}_u \cdot \mathbf{x}_u$, and so on.
When computing the dot product, it might be helpful to remember the Gauss Equations $$\begin{cases} \mathbf{x}_{uu} = \Gamma^1_{11}\mathbf{x}_u + \Gamma^{2}_{11}\mathbf{x}_v + L_{11}n \\ \mathbf{x}_{uv} = \Gamma^1_{12}\mathbf{x}_u + \Gamma^{2}_{12}\mathbf{x}_v + L_{12}n \\ \mathbf{x}_{vv} = \Gamma^1_{22}\mathbf{x}_u + \Gamma^{2}_{22}\mathbf{x}_v + L_{22}n, \\ \end{cases}$$ where the $L_{11}, L_{12}, L_{22}$ are the coefficients of the second fundamental form.