Let $M = \{ (x^\varphi, x^\theta) : x^\varphi \in [0, \pi), \thinspace x^\theta \in [0, 2\pi) \}$ be a manifold with metric $\mathrm{d}s^2 = (\sin x^\varphi)^2 (\mathrm{d} {x^\theta})^2 + (\mathrm{d} {x^\varphi})^2$ (that of an ordinary 2-sphere in spherical coordinates).
Let $x = (x^\varphi, x^\theta) \in M$ be a point on $M$ and let $v = (v^\varphi, v^\theta) \in T_xM$ be a vector on the tangent space of $M$ at $x$. Let $y = (y^\varphi, y^\theta) = \exp_x(v)$ be the point on $M$ generated by the exponential map of $v$:
It is the point at $t = 1$ on the geodesic $\gamma : t \in I \rightarrow y \in M$ such that $\gamma(0)=x$ and $\dot{\gamma}(0)=v$.
How can I find $(y^\varphi, y^\theta)$ in terms of $(x^\varphi, x^\theta)$ and $(v^\varphi, v^\theta)$, given this particular metric?
Edit
Using the metric
$$ \left( \begin{array}{cc} g_{\theta\theta} & g_{\theta\varphi} \\ g_{\varphi\theta} & g_{\varphi\varphi} \end{array} \right) = \left( \begin{array}{cc} (\sin x^\varphi)^2 & 0 \\ 0 & 1 \end{array} \right) $$
with inverse
$$ \left( \begin{array}{cc} g^{\theta\theta} & g^{\theta\varphi} \\ g^{\varphi\theta} & g^{\varphi\varphi} \end{array} \right) = \left( \begin{array}{cc} (\sin x^\varphi)^{-2} & 0 \\ 0 & 1 \end{array} \right) $$
I was able to calculate the Christoffel symbols
$$ \Gamma^a_{bc} = \frac{1}{2} g^{ad} (\partial_b g_{cd} + \partial_c g_{db} - \partial_d g_{bc}) $$
with non-zero components
$$ \Gamma^\varphi_{\theta\theta} = -\sin x^\varphi \cos x^\varphi $$ $$ \Gamma^\theta_{\theta\varphi} = \Gamma^\theta_{\varphi\theta} = (\sin x^\varphi)^{-1} \cos x^\varphi $$
From the geodesic equation
$$\frac{\partial^2 x^a}{\partial t^2}+\Gamma^a_{bc}\frac{\partial x^b}{\partial t}\frac{\partial x^c}{\partial t}=0$$
it seems I have to solve the equations
$$ \frac{\partial^2 x^\theta}{\partial t^2} = -2 \cot x^\varphi \frac{\partial x^\theta}{\partial t} \frac{\partial x^\varphi}{\partial t} $$
$$ \frac{\partial^2 x^\varphi}{\partial t^2} = \sin x^\varphi \cos x^\varphi \left(\frac{\partial x^\theta}{\partial t}\right)^2 $$
Is there a closed-form, analytic solution for these, as a function of $x(0)$, $\dot{x}(0)$, and $t$?