It is possible to solve this kind of ODE: $G''(w)+\left(\lambda^2-iw\mu \,e^{iw\mu}\right)G(w)=0$?
where $\lambda$ and $\mu$ are real-valued constants.
Motivation
I am looking for examples of Delayed Differential Equations (DDE) with closed-form solutions, and looking for them I found this question on MSE where someone ask for solving an example. By applying the Fourier Transform I found the equation of this question (hopefully I don't made any mistake), but I never tried to solve before an ODE with complex non-constant coefficients:
Are there method for solving these kind of equations with closed-form solutions?,
Wolfram-Alpha don't give any answer neither clues for what I should be looking for.
So far the only DDEs with close-form solution I have found is Wikipedia example: $$x'(t) = -x(t-1) \quad\Rightarrow x(t)=e^{t\ W_k(-1)}$$ with $W_k(t)$ the Lambert W-Function, and I don't have any clue so far about how the parameter $k$ is determined from the initial values.