I am studying Delay Differential Equations. I have an annoyingly persistent question that I haven't been able to resolve after looking at several sources. Let me provide an example to illustrate. Consider the DDE $$x^{\prime \prime}(t)+x(t-T)=0$$ with some initial history $x(t) = \phi (t)$ for $t\in [-T,0].$ The standard move is to make the ansatz that $x=e^{\lambda t}$, from which we arrive at the characteristic equation $\lambda^2 + e^{-\lambda T}=0$. This is a transcendental equation for $\lambda$ and can be solved in terms of the Lambert $W$ function, and gives countably many solutions. Let us label these solutions $\lambda_{n}$ where $n\in \mathbb{z}$. Because the equation is linear, expressions of the form $\sum_{i\in \mathbb{Z}} a_i e^{\lambda_i t}$ also solve the equation. My questions are the following.
How does one determine the coefficients $a_i$ given the initial history function $\phi$? In Erneaux's *Applied Delay Differential Equations", he says simply that the coefficients can be determined "using the Laplace Transform," but provides no details on how this might proceed. (This is in discussing his equation 2.1)
Is it known for linear DDEs that the solution set ${e^{\lambda_i t} }$ is dense in the space of solutions? More practically, I am asking if we know that all solutions are of the form $\sum_{i\in \mathbb{Z}} a_i e^{\lambda_i t}$ for appropriate choices of $a_i$, or is it possible that there are other solutions not expressible in this form?
I would appreciate any insights you have about any of the above questions, as well as a reference that does addresses any of the above in detail.
Thank you!