I'm trying to understand the working in this Wikipedia article example on DDE's, but it's not explained clearly enough and I can't understand what they're doing or why they're doing it. I've looked at answers for other problems on MathSE as well, but I feel like since I can't even understand the solution to possibly the simplest DDE, it makes me feel like I won't/can't understand the others either. I'm currently studying this myself, so I guess that is why I am slightly confused. Could anyone help explain? I'd be extremely grateful!
Stating the problem that I'm working on down here: $$x'(t) = ax(t-\tau)$$
So what I do know is that we start by working this out on smaller intervals, right? So we have $$x'(t) = ax(t-1)$$ $$x'(t) = ax(t-2)$$ and then so on, if I'm not wrong? And then we have the intervals we're working on, however I'm not sure how those progress, so please correct me if I'm stating this incorrectly. But for the first equation, we can start with $x[-1,0] = x_1$, $x[0,1] = x_2$, where $x_1$ and $x_2$ are some stated values(?), for the next one and then changing the intervals for each subsequent equation we get, right? I'm honestly just very confused and I've been looking in a lot of places but I can't find anything that explains the solution fully, so I'd really appreciate if anyone at all could help, even a little bit.
A couple of things I don't understand on the article are mostly just how they get the first equation in which $x(t) = x(0) + $ the two integrals mentioned there. Where does this form come from in the first place. I'm honestly just so confused at this point, and it is not making any sense at all :/
Thanks all for taking the time to read this though!
I'm not familiar with DDEs in particular, but it seems you should think of them as analogous to recursive sequences.
As they're continuous rather than discrete, instead of having an initial condition of $a_n=0$, say, you specify $\phi$ in an interval (because the DE only specifies values for the next chunk based on the previous one).
In essence, if you know the solution in $[n\tau, (n+1)\tau]$, the DE tells you about the next chunk along. The boundary condition is given by the previous solution as well.
In the example linked, the first integral is just the fundamental theorem of calculus: $$ x(t) = x (0) + \int_0^t x'(s) ds $$ Substituting $x'(s)=ax(s-\tau)=a\phi(s-\tau)$, where the first equality is from the DE and the second holds for $s-\tau\in[-\tau, 0]$ (matching up with the domain of integration – $0<t<\tau$ here) gives the second integral.
Hope that helps.