Find a real general solution of the following systems. (Show...

Find a real general solution of the following systems. (Show the details.)
$$\begin{aligned}
&y_{1}^{\prime}-2 y_{1}+8 y_{2}-4 y_{2}\\
&\begin{array}{l}
y_{2}^{\prime}=-4 y_{1}-10 y_{2}+2 y_{3} \\
y_{8}^{\prime}=-4 y_{1}-4 y_{2}-4 y_{z}
\end{array}
\end{aligned}$$

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Key Concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are critical for analyzing the intrinsic modes of a system. They allow the transformation of a coupled system into independent components, thereby simplifying the process of finding the general solution by decoupling the equations.

General Solution of Differential Systems

The general solution to a linear differential system is typically constructed as a linear combination of solutions corresponding to the eigenvalues and eigenvectors of the system's coefficient matrix. This fundamental concept provides a complete description of the system's dynamic behavior in terms of its independent modes.

Linear Differential Systems

This concept involves studying sets of differential equations where multiple functions interact and their derivatives depend on each other. Understanding these systems allows for the modeling and analysis of multi-dimensional dynamics in various fields such as physics, engineering, and economics.

Matrix Representation

Expressing a system of linear differential equations in matrix form consolidates multiple equations into a single compact expression. This approach enables the use of linear algebra techniques including matrix operations, which are essential for analyzing the behavior of the system.

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