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