Consider the following differential equation


$$F(cx) = F(x) + x F'(x)$$


for $c>1$.


1. Does this differential equation belong to a some well known class?
2. Is there a way to find all the solutions $F(\cdot)$ of this equation that are also cumulative distribution functions?
3. $F(x) = x^a$ for a properly chosen $a$ is a solution. Is it unique in the class of cumulative distribution functions?


P.S. It is a repost from <https://math.stackexchange.com/questions/565758/differential-equation-for-cdf>



Unusual Differential Equation for CDF
You do not tell the range of $x$. Distribution functions are usually defined on the real line, while $x^a$ on the real line is not a distribution function for any $a$.


Anyway, here is a way to solve your equation. Set $F(x)=\phi(\log x)$ and then $\log x=t$. You obtain
$$\phi(a+t)=\phi(t)+\phi'(t),$$
where $a=\log c$. This is a linear differential-difference equations, and there is a large
literature on equations of this type. The usual method is to apply Fourier transform.
It gives
$$e^{iaz}\Phi(z)=\Phi(z)+iz\Phi(z),$$
So $\Phi(z)(e^{iaz}-iz-1)=0.$
This means that $\Phi$ is a linear combination of delta-functions sitting at the roots
of the equation $e^{iaz}-iz-1$. So $\phi$ is an exponential sum, and $F$ is a linear combination of (complex) powers.


See also the discussion in:
[On equation f(z+1)-f(z)=f'(z)](https://mathoverflow.net/questions/114875/on-equation-fz1-fz-fz/114878#114878)


