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?

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