Does the definition of $\exp$ as the solution to $y'=y, y(0)=1$ allow us to actually calculate its value at a given point?

Question

Let $\exp:\Bbb R\to \Bbb R$ denote the solution of the ODE

$$ y'=y, \quad y(0)=1. $$

Say I want to calculate $\exp(x_0)$ for some given $x_0$, is there any way to do this without using the other definitions?

Answer 1

Among other things, you could get an approximate numerical value by solving the differential equation numerically.

EDIT: Methods you can use by hand or with a small amount of programming include the Euler, Improved Euler and Fourth-Order Runge-Kutta methods. Most mathematical software systems (e.g. Maple, Mathematica, Matlab etc) include more powerful methods.