# Function is equal to its own derivative [duplicate]

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
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.
• @JarredAllen Yes but that means $y'=0$ so the separation was already invalid in this special case. Additionally, $y < 0$ needs to be dealt with by premultiplying with $-1$. In the end these cases can be dealt with by allowing $\tilde c\in \mathbb R$ instead of $\tilde c > 0$ in the general solution. – AlexR Aug 25 '16 at 8:15