GR: find covariantly constant vector on a given curve

Question

2. Relevant equations

I have computed the christoffel symbols via comparing the Euler-Lagrange equations to the form expected from geodesic equation.

geodesic equation: $\ddot{x^a}+\Gamma^a_{bc}\dot{x^b}\dot{x^c}=0$

covariantly constant equation: $ V^a \nabla_a W^b = V^a (\partial_a W^b) + V^a \Gamma^b_{ac} W^c= 0 $ 1 where $V^a $ is the tangent vector to the geodesic.

I have computed the christoffel symbols as:

$\Gamma^{x}_{tt}=\frac{-1}{2x^2}$ and $ \Gamma^{t}_{tx}=\frac{-1}{2x}$

3. The attempt at a solution

From the information given $x^u=(t,1) \implies V^u=(1,0)=\delta^u_t $

Therrefore 1 non-zero equations reduces to:

$ \nabla_t W^b = (\partial_t W^b) + V^t \Gamma^b_{tc} W^c= 0 $

Using the christoffel symbols non-zerro equations further reduce to:

$\partial_t W^t - \frac{1}{2x}W^x=0$

and $ (\partial_t W^x) -\frac{W^t}{x^2}= 0 $

MY QUESTION:

so it is at this point that I am stuck. the only way I can see to proceed is to differentiate either one of the equations again wrt $t$ to get a second-order equation and then substitute in the other equation. However to then solve completely we would need 2 boundary conditions, but are only given one.

Many thanks in advance