Final answer:
The student is tasked with finding a fundamental system of solutions for a set of linear differential equations, which can be done by determining the eigenvalues and eigenvectors of the corresponding coefficient matrix we get (A - \(\lambda_i\)I)\(v_i\) = 0, where \(v_i\) is the eigenvector associated with eigenvalue \(\lambda_i\).
Step-by-step explanation:
The student's question relates to finding a fundamental system of solutions for a set of differential equations. Since the question involves a system of first order linear differential equations, we approach the task by writing the system in matrix form and solving for the eigenvalues and eigenvectors, which provide the fundamental system of solutions.
First, let's rewrite the system of differential equations in matrix form:
- \(x_1' = -x_1 - 2x_2 - x_3\)
- \(x_2' = -x_1 + x_2 + x_3\)
- \(x_3' = x_1 - x_3\)
The corresponding matrix for the coefficients is:
\( A = \begin{bmatrix} -1 & -2 & -1 \\ -1 & 1 & 1 \\ 1 & 0 & -1 \end{bmatrix} \)
To find the characteristic polynomial and hence the eigenvalues, we solve the equation det(A - \(\lambda\)I) = 0.
After determining the eigenvalues \(\lambda_i\), we find the corresponding eigenvectors by solving the system (A - \(\lambda_i\)I)\(v_i\) = 0, where \(v_i\) is the eigenvector associated with eigenvalue \(\lambda_i\). The set of eigenvectors forms the fundamental system of solutions.