I read a paper about boundary conditions and imaginary potentials. In this paper the author considers a single, non-relativistic quantum particle of mass $m > 0$ in $1$ dimension with a soft detector in finite interval $[0,L]$, $L>0$, with Schrödinger equation \begin{equation} i\hbar \frac{\partial\psi}{\partial t} = - \frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2}- iv \Theta(x)\psi(x) \end{equation} where $v>0$ is a constant and $\Theta(x)=1$ for $x\geq 0$ and $0$ otherwise. At $L>0$ he considers the Neumann boundary condition $\frac{\partial \psi_t}{\partial x}(L)=0$.
The author writes, that for $x<0$, being an eigenfunction of $-(\hbar^2/2m)\partial^2/\partial x^2$ means to solve an ODE in $x$, whose general solution has the form \begin{equation} f(x)=d_k e^{ikx}+c_ke^{-ikx}\,. \end{equation}
My question is: How did he find this eigenfunction and how do you compute an eigenfunction of a wave function with boundary condition?