Solve differential equations with an initial condition in the form of a limit as $t\rightarrow \infty$

Question

I want to solve the following differential equation

with the initial condition

This is known as the Bunch-Davies vacuum, and the above are equations (4.1) and (3.37) from this paper

Here is what I have tried so far,

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

However, I do not know how to input this initial condition in NDSolve.

Thanks!

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Please let write my equation in the new form:

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

How can solve this one?

Answer 1

After correcting few elementary syntax errors you might try this:

 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])}}

*)

with a warninhg: "Unable to resolve some of the arbitrary constants in the general solution using the given boundary conditions....". The warning indicates that the initial conditions should be chosen more carefully.

Have fun!