= Solution
In the propagated frame $T=e_0$, and the <constant sectional curvature> formula reduces <geodesic deviation> to
$$
\ddot y^0=0,
\qquad
\ddot y^i=K y^i.
$$
The stated temporal initial data give $y^0(t)=0$. Writing $\omega=\sqrt{-K}$ for $K<0$ and $\kappa=\sqrt K$ for $K>0$, the spatial displacement is
$$
\boxed{
y^i(t)=
\begin{cases}
Y^i\cos(\omega t)+(V^i/\omega)\sin(\omega t),&K<0,\\
Y^i+V^it,&K=0,\\
Y^i\cosh(\kappa t)+(V^i/\kappa)\sinh(\kappa t),&K>0.
\end{cases}}
$$
These formulas are valid to first order in the initial separation and relative velocity.
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