I have the following system of complex differential equations $$ \frac{dA}{dz} = K_{ab}Be^{-i\Delta z}$$ $$ \frac{dB}{dz} = K_{ba}Ae^{i\Delta z}$$ where A,B,$K_{ab}$,$K_{ba}$ are complex, with initial conditions A(0)=0, B(0) = $B_{0}$ and $K_{ab} = -K_{ba}^*$. The first thing I do is take derivative of first equation with respect to z. Then I have in simplified form $$\frac{d^2A}{dz^2} = K_{ab}K_{ba}A-i\Delta \frac{dA}{dz} $$ $$\frac{d^2A}{dz^2} + i\Delta \frac{dA}{dz} + |K_{ab}|^2A = 0 $$
I couldn't proceed with success, assuming this second order equation has the solution in the form of $e^{sz}$. The discriminant of characteristic polynomial is less than zero and roots are not conjugate of each other. I have looked Arfken Mathematical Methods but I couldn't see the solution. Thanks in advance I am open to any textbook advice on complex differential equations
$e^{isz}$ solves the differential equation but I am not satisfied, since I found it by trial and error.