Solutions for complex conjugate roots

Statement

Let $a_0, \dots, a_{n-1}$ be real numbers, and let $p$ be the characteristic polynomial of the equation $y^{(n)} + a_{n-1} y^{(n-1)} + \dots + a_0 y = 0$. Let $\alpha + i \beta$ be a root of $p$ of multiplicity $m$, with $\alpha, \beta \in \mathbb{R}$ and $\beta \ne 0$. Then $\alpha - i \beta$ is also a root of multiplicity $m$, and the $2m$ functions $$x^k e^{\alpha x} \cos(\beta x), \quad x^k e^{\alpha x} \sin(\beta x), \qquad k = 0, \dots, m - 1,$$ are linearly independent real solutions of the equation on $\mathbb{R}$.