= Solution
For an affine parameter, the <geodesic equation> is
$$
\boxed{\ddot x^a+\Gamma^a_{bc}\dot x^b\dot x^c=0.}
$$
Vary the energy action with endpoint-fixed variations. Its Euler-Lagrange equation is
$$
2g_{ab}\ddot x^b+2\partial_cg_{ab}\dot x^c\dot x^b
-\partial_ag_{bc}\dot x^b\dot x^c=0.
$$
Multiplying by $g^{da}/2$ and symmetrizing the velocity indices determines the <Levi-Civita connection>:
$$
\boxed{\Gamma^d_{bc}=\frac12g^{da}
(\partial_bg_{ac}+\partial_cg_{ab}-\partial_ag_{bc}).}
$$
This determines the torsion-free metric connection; an antisymmetric-in-$b,c$ addition could not be detected by the geodesic energy variation alone.
For the displayed metric on $t>0$, the nonzero coefficients are $\Gamma^x_{xt}=\Gamma^x_{tx}=-1/t$ and $\Gamma^t_{xx}=\Gamma^t_{tt}=-1/t$. Consequently
$$
\ddot x-2\dot x\dot t/t=0,\qquad
\ddot t-(\dot x^2+\dot t^2)/t=0.
$$
For a timelike curve choose <proper time> so $(\dot x^2-\dot t^2)/t^2=-1$. The first equation integrates to $\dot x=pt^2$, and then $\dot t^2=t^2+p^2t^4$. If $p\ne0$, dividing and integrating gives
$$
x=x_c\pm\sqrt{t^2+p^{-2}},\qquad
\boxed{t^2=x^2+\alpha x+\beta,\quad
\alpha=-2x_c,\quad\beta=x_c^2-p^{-2}.}
$$
The timelike condition is $\alpha^2-4\beta=4p^{-2}>0$, with the appropriate branch in the coordinate patch. Conversely these curves with that condition obey the first integrals and admit a timelike affine parametrization.
There is an omitted family in the printed claim: if $p=0$, then \b[$x=x_c$ is a timelike vertical geodesic], with $t=t_*e^{\pm\tau}$ in <proper time>. No fixed finite $\alpha,\beta$ can represent all points of such a vertical curve by the displayed quadratic. The quadratic therefore describes the nonvertical <timelike geodesics>; the vertical family must be included for completeness.
Together the two families describe the <timelike geodesics of an inverse-time conformal metric>.
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