Eigenfunction of a wave function with boundary condition

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}
ihbar frac{partialpsi}{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 $xgeq 0$ 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?