Definition:Exponential Function/Real/Differential Equation

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Let $\exp: \R \to \R_{>0}$ denote the (real) exponential function.

The exponential function can be defined as the unique solution $y = f \left({x}\right)$ to the first order ODE:

$\dfrac {\d y} {\d x} = y$

satisfying the initial condition $f \left({0}\right) = 1$.

That is, the defining property of $\exp$ is that it is its own derivative.

The number $\exp x$ is called the exponential of $x$.