TITLE: Find $f:\mathbb R\to\mathbb R$ such that $f(x)=f'(x)$.
QUESTION [6 upvotes]: Find $f:\mathbb R\to\mathbb R$ such that $f(x)=f'(x)$.

I know that $f(x)$ can be $e^x$, but how to solve this mathematically? Are there any other solutions? Thank you.

REPLY [3 votes]: Simple elementary differential equation. Put $\,y=f(x)\,$ , then::
$$f'(x)=f(x)\iff \frac{dy}{dx}=y\implies \int\frac{dy}{y}=\int dx\implies$$
$$\implies \log y=x+c\implies y=e^{x+c}=Ke^x\;,\;\;K:=e^c=\text{ a constant}$$