1) What type of differential equation is: $$ \frac{\partial A(t)}{\partial t}=a*k(t)^{2}*A(t)+i*b*k(t)*A(t) $$
2) Can I solve it by separation of variables? If not, how to do so?
3) Does it change anything in that the second part on the RHS is imaginary or schould I just treat it like a normal constant?
Thank you for help!
Your equation can be written as $$ \frac{dA}{dt} = [ak^2(t) + ibk(t)] A. $$ You can see that it can indeed be solved by separation of variables as long as you known $k(t)$ (otherwise, you can write the solution for $A$ in terms of $k$).
After you solve it, you will get $A$ as the exponential of some integrals. If your function $k(t)$ is real valued, your solution will have a real part and a imaginary part, that you will be able to obtain using the Euler's formula. Before that, you can deal with $i$ as a mere constant.