TITLE: Function is equal to its own derivative
QUESTION [7 upvotes]: We all know that derivative of $e^x$ is $e^x$. Is exponential function only function that has such property? If yes how to prove that there are no other functions. If no, what are other functions? Help me please

REPLY [12 votes]: You seek to solve the ODE $y'=y$ for arbitrary boundary conditions. This is separable and yields
$$1 = \frac y{y'}$$
Integration gives
$$x+c = \ln(y(x))$$
or
$$y(x)=e^{x+c} = e^c e^x = \tilde c e^x$$
The uniqueness is guaranteed by Picard-Lindelöf.