\begin{definition}[Definition:Exponential Function/Real/Differential Equation]
Let $\exp: \R \to \R_{>0}$ denote the '''(real) exponential function'''.
The '''exponential function''' can be defined as the unique solution $y = \map f x$ to the first order ODE:
:$\dfrac {\d y} {\d x} = y$
satisfying the initial condition $\map f 0 = 1$.
That is, the defining property of $\exp$ is that it is its own derivative.
The number $\exp x$ is called the '''exponential of $x$'''.
\end{definition}