Solutions for repeated roots

Statement

Let $a_0, \dots, a_{n-1}$ be real numbers, and let $p(r) = r^n + a_{n-1} r^{n-1} + \dots + a_0$ be the characteristic polynomial of the equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = 0$. Let $r$ be a real root of $p$ of multiplicity $m \ge 1$. Then the $m$ functions $e^{r x}, x e^{r x}, \dots, x^{m-1} e^{r x}$ are linearly independent solutions of the equation on $\mathbb{R}$. If all the roots of $p$ are real, the union of these families over the distinct roots of $p$ is a fundamental system.