Solutions for distinct real roots

Statement

Let $a_0, \dots, a_{n-1}$ be real numbers, and suppose that the characteristic polynomial $p(r) = r^n + a_{n-1} r^{n-1} + \dots + a_0$ has $n$ distinct real roots $r_1, \dots, r_n$. Then $\left(e^{r_1 x}, \dots, e^{r_n x}\right)$ is a fundamental system of the equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = 0$ on $\mathbb{R}$. So the solutions are exactly the functions $y = c_1 e^{r_1 x} + \dots + c_n e^{r_n x}$ with real constants $c_1, \dots, c_n$.