91Ó°ÊÓ

The coefficient matrix is a fundamental concept when dealing with systems of linear equations, including those formed by linear differential equations. It organizes the coefficients of each variable into a compact matrix form. In the system of differential equations provided:

• \( x_{1}'=2x_{1}+x_{2}-x_{3} \)
• \( x_{2}'=-4x_{1}-3x_{2}-x_{3} \)
• \( x_{3}'=4x_{1}+4x_{2}+2x_{3} \)

the coefficient matrix \( A \) consolidates all the coefficients of \( x_1, x_2, \) and \( x_3 \) into:\[A = \begin{pmatrix} 2 & 1 & -1 \ -4 & -3 & -1 \ 4 & 4 & 2 \end{pmatrix}\]This matrix directly expresses how the current state of the system influences its time evolution, described by \( \mathbf{x}' = A\mathbf{x} \). This formulation simplifies the manipulation and solution of the system.

The characteristic polynomial is a crucial element for finding eigenvalues, which provide insight into the behavior of a system. To derive the characteristic polynomial, compute the determinant of \( A - \lambda I \), where \( \lambda \) is a scalar and \( I \) is the identity matrix of the same size as \( A \). For our coefficient matrix:\[A = \begin{pmatrix} 2 & 1 & -1 \ -4 & -3 & -1 \ 4 & 4 & 2 \end{pmatrix}\]we have:\[A - \lambda I = \begin{pmatrix} 2-\lambda & 1 & -1 \ -4 & -3-\lambda & -1 \ 4 & 4 & 2-\lambda \end{pmatrix}\]The determinant of this matrix leads to the polynomial:\[det(A - \lambda I) = - (\lambda + 3)(\lambda - 3)^2\]Solving the equation \( \,det(A - \lambda I) = 0 \,\) gives the eigenvalues of the system, helping us understand how different modes of the system decay or grow over time.

Linear differential equations involve derivatives of one or more variables where each term is either a constant or the product of a constant and a single variable. These equations often model real-world systems such as electrical circuits or population dynamics. In our exercise, three coupled linear differential equations describe a dynamic system with variables \( x_1, x_2, \) and \( x_3 \):

• \( x_{1}'=2x_{1}+x_{2}-x_{3} \)
• \( x_{2}'=-4x_{1}-3x_{2}-x_{3} \)
• \( x_{3}'=4x_{1}+4x_{2}+2x_{3} \)

These equations show how each variable's rate of change depends linearly on the others, forming the basis for many physical and engineering systems. Solving these equations requires finding eigenvalues and eigenvectors, which unravel the system's general solution.

The general solution of a system of linear differential equations describes how the system evolves over time. It captures the complete set of potential states the system could experience. Using the eigenvalue method, the solution is expressed as a linear combination of terms involving eigenvectors and exponential functions of eigenvalues.For the given system with eigenvalues \( \lambda_1 = -3 \) and \( \lambda_2 = \lambda_3 = 3 \), and their respective eigenvectors, the solution can be written as:\[\mathbf{x}(t) = c_1\begin{pmatrix} 1 \ -1 \ 0 \end{pmatrix}e^{-3t} + c_2\begin{pmatrix} 1 \ -2 \ 4 \end{pmatrix}e^{3t} + c_3\begin{pmatrix} 3 \ -4 \ 8 \end{pmatrix}e^{3t}\]Here, \( c_1, c_2, \) and \( c_3 \) are constants set by initial conditions. The exponential terms dictate how quickly each state of the system will increase or decrease over time, providing a comprehensive view of the entire system's behavior.