differential equations - Using DSolve with a boundary condition at -Infinity

I would like to solve a simple 2nd-order ODE with one of the boundary conditions defined at $ -\infty $. The ODE I am looking to solve is:

$$ w''(z)-2i\pi^2w(z)=0 $$

with the corresponding boundary conditions:

$$ w(z=-\infty)=0, \; w'(z=0)=0+i\dfrac{\tau_{0}}{\mu}. $$

My attempt at a solution using DSolve is as follows:

DSolve[{-2 I \[Pi]^2 w[z] + (w^\[Prime]\[Prime])[z] == 0, 
w[-Infinity] == 0, w'[0] == 0 + I Subscript[\[Tau], 0]/\[Mu]}, w[z],z]

but I only get an empty set of curly brackets as an output. I checked the rest of my snipet of code without the w[-Infinity]==0 boundary condition, and that works as expected; therefore, I know that this is a problem with the boundary condition at $z=-\infty$. I am looking for methods with which I can solve simple ODE's with boundary conditions at infinity, and any help would be greatly appreciated.

Answer

This is the solution of your equation without the boundary conditions:

sol = DSolve[-2 I \[Pi]^2 w[z] + w''[z] == 0, w[z], z] // ExpToTrig //
   ComplexExpand

(*  {{w[z] -> 
   C[1] Cos[\[Pi] z] Cosh[\[Pi] z] + C[2] Cos[\[Pi] z] Cosh[\[Pi] z] +
     C[1] Cos[\[Pi] z] Sinh[\[Pi] z] - C[2] Cos[\[Pi] z] Sinh[\[Pi] z] + 

    I (C[1] Cosh[\[Pi] z] Sin[\[Pi] z] - C[2] Cosh[\[Pi] z] Sin[\[Pi] z] + 
       C[1] Sin[\[Pi] z] Sinh[\[Pi] z] + C[2] Sin[\[Pi] z] Sinh[\[Pi] z])}}  *)

Now let us take its limit at z->-Infinity:

Limit[w[z] /. sol, z -> -\[Infinity]]

(* {ComplexInfinity}  *)

Let us now try this limit at C[1]=0 and C[2]=0:

Limit[w[z] /. sol /. {C[1] -> 0}, z -> -\[Infinity]]


(* ComplexInfinity *)

Limit[w[z] /. sol /. {C[2] -> 0}, z -> -\[Infinity]]

(*  0  *)

The latter gives us what we need, therefore, C[2]=0.

Let us now implement the second boundary condition:

    Solve[(D[(w[z] /. sol /. C[2] -> 0), z] /. z -> 0) == I*t/m, C[1]]


(*  {{C[1] -> ((1/2 + I/2) t)/(m \[Pi])}}  *)

Done. Have fun!