Homogeneous linear differential equation of order n with constant coefficients

Homogeneous linear differential equation of order n with constant coefficients

$$\frac{d^{n}y}{d{t}^n}+a_{n-1}\frac{d^{n-1}y}{d{t}^{n-1}}+\cdots +a_1\frac{dy}{dt}+a_{0}y=0$$

Because the operator is linear mapping, the solution space is a vector space

suppose $e^{\lambda t}$ is a solution, then

$$\begin{array}{rl}\frac{d^n}{d{t}^n}{e}^{\lambda t}+a_{n-1}\frac{d^{n-1}}{d{t}^{n-1}}{e}^{\lambda t}+\cdots +a_1\frac{d}{dt}{e}^{\lambda t}+a_0{e}^{\lambda t}& =0\\ \left(\frac{d^n}{d{t}^n}+a_{n-1}\frac{d^{n-1}}{d{t}^{n-1}}+\cdots +a_1\frac{d}{dt}+a_0\right)e^{\lambda t}& =0\\ {\lambda }^n{e}^{\lambda t}+a_{n-1}{\lambda }^{n-1}{e}^{\lambda t}+\cdots +a_1\lambda e^{\lambda t}+a_0{e}^{\lambda t}& =0\\ \left({\lambda }^n+a_{n-1}{\lambda }^{n-1}+\cdots +a_1\lambda +a_0\right)e^{\lambda t}& =0\\ \left({\lambda }^n+a_{n-1}{\lambda }^{n-1}+\cdots +a_1\lambda +a_0\right)& =0\end{array}$$

the characteristic equation is

$$p(\lambda )={\lambda }^n+a_{n-1}{\lambda }^{n-1}+\cdots +a_1\lambda +a_0=0$$

there must be $m$ distinct roots, ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_m$, the multiplicity of the root is $k_1,k_2,\cdots ,k_m$

the root ${\lambda }_i$ has multiplicity $k_i$, then the solution is

$$y=\sum _{i=1}^m\sum _{j=0}^{k_i-1}{c}_{ij}{t}^j{e}^{{\lambda }_{i}t}$$