Let \( \mathbf{y}^{\prime}=A \mathbf{y} \) be a system of differential equations where \( A=\left[\begin{array}{cc}3 & -1 \\ 0 & 3\end{array}\right] \) The matrix has spectrum \( \lambda(A)=\{3\} \) and \( \mathbf{c}=\left[\begin{array}{l}1 \\ 0\end{array}\right] \) is an eigenvector of \( A \) corresponding to \( r=3 \) The vector \( \mathbf{d}=\left[\begin{array}{c}2 \\ -1\end{array}\right] \) satisfies the equation \( (A-r I) \mathbf{d}=\mathbf{c} \) What is the general solution to the system of differential equations? \[ \left[\begin{array}{l}y_{1} \\ y_{2}\end{array}\right]=k_{1} e \text { Ex6 } t\left[\begin{array}{l}\text { Ex: } 6 \\ \square\end{array}\right]+k_{2}(t e \]

To tackle the given system of differential equations, we start by identifying the general form of the solution for a repeated eigenvalue. Since \( \lambda = 3\) is the only eigenvalue, we have one eigenvector \( \mathbf{c} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) and, due to the Jordan form, we can derive a generalized eigenvector \( \mathbf{d} = \begin{bmatrix} 2 \\ -1 \end{bmatrix} \). The general solution takes the form: \[ \mathbf{y}(t) = e^{3t} \left( k_1 \begin{bmatrix} 1 \\ 0 \end{bmatrix} + k_2 \left( t \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} 2 \\ -1 \end{bmatrix} \right) \right) = k_1 e^{3t} \begin{bmatrix} 1 \\ 0 \end{bmatrix} + k_2 e^{3t} \left( t \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} 2 \\ -1 \end{bmatrix} \right) \] This indicates that the solutions compound over time, creating a rich behavior in the dynamic relationships defined by the system. **The general solution to the system of differential equations is:** \[ \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} = k_1 e^{3t} \begin{bmatrix} 1 \\ 0 \end{bmatrix} + k_2 e^{3t} \left( t \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} 2 \\ -1 \end{bmatrix} \right) \] --- When dealing with differential equations, it's useful to consider the implications of the eigenvalues and the associated eigenvectors. In this case, since we have a repeated eigenvalue, the presence of the generalized eigenvector indicates that the solution diverges from a simple exponential growth to a more complex, polynomial-exponential combination. This approach is common in understanding how systems might behave under perturbations; it's the intricacies of repeated eigenvalues that often lead to systems displaying diverse dynamic characteristics beyond their linear components.