$sin(x)$ and $cos(x)$ both satisfy the Delay Differential Equation
$f'(x) = f(x+a)$ with $a = \pi/2$
I have consulted other questions on this forum and understood there are more solutions to the Delay Differential Equation $f'(x) = f(x+a)$. But are there any others that oscillate, and is the period always $4a$?
For this particular solution "$2\pi$" and thus "$4a$" emerges as the period $w$. Obviously many solutions just keep going up or down, related to the real exponential function $e^{cx}$, but some solutions seem to "oscillate".
A function like $\sin(5x)$ then has a period which is 5 times smaller but still satisfies $w = 4a$.
I am trying to understand what this means in terms of "the meaning of $\pi$". Is "$\pi$" special in the sense that it's the only period where you can construct a function that oscillates in a certain manner?
I also have no idea how this relates to the second order derivative of $f$, which for $\sin$ obviously has a relationship with itself, in the form of a simple second order differential equation.
I spent time with all answers regarding delay differentials and the Wikipedia page for them, but couldn't really get any further in understanding whether or not this insight reveals something about pi - something I realise that will most likely be tautological to its geometric meaning.
Taking for convenience
$$ f'(t) = f(t+a) $$
and transforming Laplace we have
$$ sF(s)-e^{a s}F(s) = f(0) $$
or
$$ F(s) = \frac{f(0)}{s-e^{as}} $$
now if $s_k = x_k + j y_k$ are such that $s_k - e^{a s_k} = 0$ we have
$$ f(t) = f(0)\sum_{k=0}^{\infty}\left(\alpha_k\sin (y_k t)+\beta_k \cos (y_k t)\right)e^{x_k t} $$
where $\alpha_k,\ \beta_k$ are coefficients of the residue expansion for $\frac{1}{s-e^{as}}$.
The zeroes for $s-e^{a s}=0$ can be located by solving
$$ \cases{ x-e^{a x}\cos(a y) = 0\\ y-e^{a x}\sin(a y) = 0 } $$
The zeroes are contained also in the trace of
$$ x^2+y^2 = e^{2a x} $$
for $a = \frac{\pi}{2}$ the left-most zero is located at
$$ 0^2+y^2 = 1\Rightarrow \alpha_0\cos t+\beta_0 \sin t $$
It is the first expansion pole, and the only pure periodic term. The following terms are exponentially weighted and then are not purely periodic.
Follow a plot showing in blue the locus for $x-e^{a x}\cos(a y) = 0$ in red the locus for $y-e^{a x}\sin(a y) = 0$ and in black the locus $x^2+y^2 = e^{2a x}$
This graphic was made for $a = \frac{\pi}{2}$
NOTE
The parametric values $a = (4k+1)\frac{\pi}{2}$ for $k = 0, 1, 2,\cdots,$ are the values such that there exists a pole which gives pure oscillations.