Example 1. Solve the system

$$
\left\{\begin{array}{l}
\frac{d x}{d t}=3 x-y+z \\
\frac{d y}{d t}=-x+5 y-z \\
\frac{d z}{d t}=x-y+3 z
\end{array}\right.
$$
Solution. We form the characteristic equation

$$
\left|\begin{array}{ccc}
3-r & -1 & 1 \\
-1 & 5-r & -1 \\
1 & -1 & 3-r
\end{array}\right|=0
$$

or $r^{3}-11 r^{2}+36 r-36=0$.

The roots $r_{1}=2, r_{2}=3, r_{3}=6$ correspond to the numbers

$$
\begin{array}{lll}
\lambda_{1}=1, & \mu_{1}=0, & \nu_{1}=-1 \\
\lambda_{2}=1, & \mu_{2}=1, & \nu_{2}=1 \\
\lambda_{3}=1, & \mu_{3}=-2, & \nu_{3}=1
\end{array}
$$

We write down the particular solutions

$$
\begin{array}{lll}
x_{1}=e^{3 t}, & y_{1}=0, & z_{1}=-e^{2 t}, \\
x_{2}=e^{3 t}, & y_{2}=e^{3 t}, & z_{1}=e^{3 t}, \\
x_{3}=e^{t t}, & y_{3}=-2 e^{6 t}, & z_{3}=e^{t t_{3}}
$$

The general solution of the system:

$$
\begin{aligned}
& x=C_{1} e^{2 t}+C_{2} e^{3 t}+C_{3} e^{6 t} \\
& y=C_{2} e^{3 t}-2 C_{3} e^{6 t} \\
& z=-C_{1} e^{2 t}+C_{2} e^{3 t}+C_{3} e^{6 t}
\end{aligned}
$$

B. Now consider the case when the roots of the characteristic equation are complex.