eq = {y''[t] + (k^2 - 2/t) y[t] == 0, y[-∞] == Exp[I*k*t], y'[-∞] == 0};
NDSolve[eq, y[t], t]

eq = {y''[t] + (k^2 - 2/t) y[t] == 0, y[-[Infinity]] == Exp[I*k*t], y'[-      [Infinity]] == 0};
 NDSolve[eq, y[t], t]

eq = {y''[t] + (k^2 - 2/t) y[t] == 0, y[0] == Exp[I*k*t], y'[0] == 0};
DSolve[eq, y[t], t]

(*
{{y[t] -> -E^(-Sqrt[k^2] t + (-Sqrt[-k^2] + Sqrt[k^2]) t) t C[
     1] (Hypergeometric1F1[
        1 + (-4 Sqrt[k^2] - 2 (-2 - 2 Sqrt[k^2]))/(4 Sqrt[-k^2]), 2,
        2 Sqrt[-k^2] t] HypergeometricU[
        1 + (-4 Sqrt[k^2] - 2 (-2 - 2 Sqrt[k^2]))/(4 Sqrt[-k^2]), 2,
        0] - HypergeometricU[
       1 + (-4 Sqrt[k^2] - 2 (-2 - 2 Sqrt[k^2]))/(4 Sqrt[-k^2]), 2,
       2 Sqrt[-k^2] t])}}

*)