I am reading the paper Do dark matter axions form a condensate with long-range correlation?.
Let $$\psi = A + iB \tag{1}$$ Performing a Fourier Transform, you get $A_k$ and $B_k$.
It is found using a physical equation that $\psi$ satisfies that $A_k$ and $B_k$ obey $$\frac{\mathrm{d}}{\mathrm{d}t}\begin{bmatrix} A_k\\ B_k \end{bmatrix} = \begin{bmatrix}0 & \frac{k^2}{2m} \\ -\frac{k^2}{2m}-\frac{\lambda n_0}{4m^2} & 0\end{bmatrix}\begin{bmatrix}A_k \\ B_k\end{bmatrix} \tag{2}$$
Defining $$\kappa_k = \frac{k^2}{2m}+\frac{\lambda n_0}{4m^2} \tag{3}$$
The paper says that the solution to eqn(2) for $\kappa_k<0$ is
$$\psi_k = c_1(\gamma_k - i\kappa_k)e^{\gamma_kt} + c_2(\gamma_k + i\kappa_k)e^{-\gamma_kt} \tag{4}$$ where $c_1$ and $c_2$ are arbitrary real constants and $\pm\gamma_k$ are the eigen vlaues of the above matrix, $$\gamma_k = \frac{k}{\sqrt{2m}}\sqrt{-\kappa_k}$$
My question is how can I get equation (4) from equation (2)?