Let $c\in\mathbb{R}$ such that $|c|>1$. Does there exist functions $f\in\mathscr{C} ^\infty(\mathbb{R})$ such that $\forall x\in\mathbb{R}, \; f'(x) =f(cx)$ and $f\neq 0$ ? If yes, can we find all those functions ?
I have tried to search analytic such $f$, but the condition $|c|>1$ tells that no such functions exist.
I have tried to work with the Taylor expansion of a solution if there exists one...without success.
Thank you for your help.