A Functional Differential Equation: $f^\prime(x) =\frac{f(2x)}{2f(x)}$

Question

I was having a play with some trig. identities and noticed the following:

$$\cos{x}=\frac{\sin{2x}}{2\sin{x}}.\tag{1}$$

Now, $\cos{x} = \frac{d}{dx}\sin{x}$ so I made the following analogous differential equation:

$$f^\prime(x) =\frac{f(2x)}{2f(x)} \tag{2}$$

I have not seen a differential equation which relates a function's derivative to a change in its argument, so I was wondering whether anyone knew what these were called?

Somewhat predictably, $f_1(x)=\sin{x}$ is not the only solution, I found that $f_2(x)=A\sin(\omega x)$ where $a\omega=1$ is also a solution. I then guessed another solution, $e^{\lambda x}$, and found that $f_3(x)=e^{\frac{1}{2}x}$ is also a solution.

My main questions are:

What are these type of equations called? and are there any other solutions to this one?

Thanks for reading.

Answer 1

To use "Picard's method" might work. You have $$ y(x)=y_0+\int_{x_0}^x \frac{y(2t)}{y(t)} \, dt $$

Define $$ y_{k+1}=y_0+\int_{x_0}^x \frac{y_k(2t)}{y_k(t)}\,dt $$ and calculate $\lim_{k\rightarrow \infty} y_k$. I do not know if the sequence converges.