Example 7. Solve the system $\left\{\begin{array}{l}x^{\prime}=3 x-y+z, \\ y^{\prime}=-x+5 y-z, \\ z^{\prime}=x-y+3 z .\end{array}\right.$
Solution. Let's provide a brief schematic solution to reinforce the methodology of eigenvalues and eigenvectors.

$$
A=\left(\begin{array}{ccc}
3 & -1 & 1 \\
-1 & 5 & -1 \\
1 & -1 & 3
\end{array}\right)
$$

1) Characteristic equation of the matrix

$A=\left|\begin{array}{ccc}3-\lambda & -1 & 1 \\ -1 & 5-\lambda & -1 \\ 1 & -1 & 3-\lambda\end{array}\right|=0$, or $\lambda^{3}-11 \lambda^{2}+36 \lambda-36=0$, has three simple roots: $\lambda_{1}=2, \lambda_{2}=3, \lambda_{3}=6$.

2) For each eigenvalue, find an eigenvector of the matrix $A$ and the corresponding particular solution of the system.

a) For $\lambda_{1}=2$, we can choose the eigenvector ( $-1,0,1$ ) and the corresponding particular solution $\left(x_{1}, y_{1}, z_{1}\right)=\left(-e^{2 t}, 0, e^{2 t}\right)$.

b) For $\lambda_{2}=3$, we can choose the eigenvector ( $1,1,1$ ) and the corresponding particular solution ( $\left.x_{2}, y_{2}, z_{2}\right)=\left(e^{3 t}, e^{3 t}, e^{3 t}\right)$.

c) For $\lambda_{3}=6$, we can choose the eigenvector $(-1,2,-1)$ and the corresponding particular solution $\left(x_{3}, y_{3}, z_{3}\right)=\left(-e^{6 t}, 2 e^{6 t},-e^{6 t}\right)$.

3) Form the general solution of the system as a linear combination of the obtained particular solutions.

![](https://cdn.mathpix.com/cropped/2024_05_22_446938f1e4808e7d6e07g-107.jpg?height=176&width=740&top_left_y=823&top_left_x=1406)

For solving systems of differential equations of a more general form, it is necessary to use more sophisticated methods of their analysis and combination.