Any periodic function which contains a scaled or translated version of its own derivative, for example sine or cosine , or any finite or infinite sum of multiple periodic functions which also yields a periodic function which is its own derivative, can be expressed in the format which you are asking for. (Assuming that you are considering the sine solution you listed as a non-trivial solution)

For example, for $k=1$, you can transform the domain and range of the function $y=sin(x)$ to $y=2\pi sin(x/{(2\pi)})$, or for arbitrary $k$, $$y=k sin(\frac{2\pi x}{k})$$

You can create a similar function for cosine by adding a phase shift to the domain. So for arbitary $k$, you could use sine or cosine with the domain scaled and translated and the range scaled as necessary to get a solution of the format you'd like.

$$ y = c g(\frac{a x}{k}+b) $$

If you are looking for a non-periodic trivial solution, then it's a different story and answer, delayed differential equations, as pointed out by Denis Serre above.

Any periodic function which contains a scaled or translated version of its own derivative, for example sine or cosine , or any finite or infinite sum of multiple periodic functions which also yields a periodic function which is its own derivative, can be expressed in the format which you are asking for. (Assuming that you are considering the sine solution you listed as a non-trivial solution)

For example, for $k=1$, you can transform the domain and range of the function $y=sin(x)$ to $y=2\pi sin(2\pi x -\pi/2)$, or for arbitrary $k$, $$y=k sin(\frac{2\pi x}{k} - \pi/2)$$

You can create a similar function for cosine by adding a different phase shift to the domain. So for arbitary $k$, you could use sine or cosine with the domain scaled and translated and the range scaled as necessary to get a solution of the format you'd like.

$$ y = c g(\frac{a x}{k}+b) $$

If you are looking for a non-periodic trivial solution, then it's a different story and answer, delayed differential equations, as pointed out by Denis Serre above.

Any periodic function, for example sine or cosine or tangent, or any finite or infinite sum of multiple periodic functions which also yields a periodic function, can be expressed in the format which you are asking for. (Assuming that you are considering the sine solution you listed as a non-trivial solution)

For example, for $k=1$, you can transform the domain of the function $y=sin(x)$ to $y=sin(x/{(2\pi)})$, or for arbitrary $k$, $$y=sin(\frac{2\pi x}{k})$$

For any arbitrary function $g(x)$ which is periodic with period length $a$, dividing the domain by the $k$ desired and multiplying by period length will yield such a function.

$$ y = g(\frac{a x}{k}) $$

If you are looking for a non-periodic trivial solution, then it's a different story and answer, differential delay equations, as pointed out by Denis Serre above.

Any periodic function which contains a scaled or translated version of its own derivative, for example sine or cosine , or any finite or infinite sum of multiple periodic functions which also yields a periodic function which is its own derivative, can be expressed in the format which you are asking for. (Assuming that you are considering the sine solution you listed as a non-trivial solution)

For example, for $k=1$, you can transform the domain and range of the function $y=sin(x)$ to $y=2\pi sin(x/{(2\pi)})$, or for arbitrary $k$, $$y=k sin(\frac{2\pi x}{k})$$

You can create a similar function for cosine by adding a phase shift to the domain. So for arbitary $k$, you could use sine or cosine with the domain scaled and translated and the range scaled as necessary to get a solution of the format you'd like.

$$ y = c g(\frac{a x}{k}+b) $$

If you are looking for a non-periodic trivial solution, then it's a different story and answer, delayed differential equations, as pointed out by Denis Serre above.