TITLE: Why can we substitute the exponential function while deriving the characteristic equation
QUESTION [6 upvotes]: I'm studying the way characteristic equation works. The derivation, according to Wikipedia, follows:
We have a function $y(x)$ and an equation
$$a_n y^{(n)} + \cdots + a_1 y' + a_0 y = 0.$$
Then we substitute $$y = e^{rx}.$$
Why can we make such a substitution? I suspect we can't represent any function as $e^{rx}$. Besides it is said that $\forall z: e^{z} \neq 0$ (can you point me to a proof too), so if $\exists x_0: y(x_0) = 0$ then we can't represent this value in this substitution.

REPLY [3 votes]: I would refer to that (setting $y= e^{rx}$) as "an educated guess".