Homogeneous n-th order ODE

#math

$$a_{0}y^{(n)} + a_{1}y^{(n-1)} + \dots+a_{n-1}y^{(1)} + a_{n}y = 0 \tag{4}$$

where the superscripts denote $n$-th differential wrt $x$, then the characteristic equation is $$a_{0}m^n + a_{1}m^{n-1} \dots+a_{n-1}m + a_{n} = 0 \tag{5}$$and it gives an idea of the roots.

The fundamental set of solutions $\mathcal{B}$ (basis) of $(4)$ are given by the following set of rules

1. Every distinct root $m_i$ contributes one solution to the basis, $e^{m_{i}x}$
2. Every root repeated $k$ times contributes $k$ solutions to the basis, $$e^{mx},\ \ xe^{mx},\ \ x^2e^{mx},\ \ x^3e^{mx},\ \dots \ x^{k-1}e^{mx}$$
3. Each individual complex conjugate $\alpha \pm i\beta$ will contribute two solution, $e^{\alpha x}\sin (\beta x)$ and $e^{\alpha x}\cos (\beta x)$.
4. Each repeated complex conjugate ($\beta \ne 0$) repeated $k$ times, contribute $2k$ LI solutions in the same fashion as rule 2. $e^{\alpha x}\sin (\beta x)$ and $e^{\alpha x}\cos (\beta x)$; $x{e}^{\alpha x}\sin (\beta x)$ and $x{e}^{\alpha x}\cos (\beta x)$; $x^2{e}^{\alpha x}\sin (\beta x)$ and $x^2{e}^{\alpha x}\cos (\beta x)$; $\dots$; $x^{k-1}{e}^{\alpha x}\sin (\beta x)$ and $x^{k-1}{e}^{\alpha x}\cos (\beta x)$.

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