Final Answer:
For a system of differential equations \( \mathbf{x}' = \mathbf{Ax} \) with a 2x2 matrix \( \mathbf{A} \), the solution \( \mathbf{x}(t) \) can be expressed as \( \mathbf{x}(t) = \mathbf{P}e^{\mathbf{Dt}}\mathbf{P}^{-1} \), where \( \mathbf{P} \) is the matrix of eigenvectors of \( \mathbf{A} \) and \( \mathbf{D} \) is the diagonal matrix of eigenvalues.
Step-by-step explanation:
In this system of differential equations, \( \mathbf{A} \) represents a 2x2 matrix, and \( \mathbf{x} = [x_1, x_2] \) is the vector of dependent variables. The solution to the system involves finding the eigenvalues and eigenvectors of \( \mathbf{A} \). Let \( \lambda_1 \) and \( \lambda_2 \) be the eigenvalues, and \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) be the corresponding eigenvectors.
The matrix \( \mathbf{P} \) is formed by arranging the eigenvectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) as columns. The matrix \( \mathbf{D} \) is a diagonal matrix with the eigenvalues \( \lambda_1 \) and \( \lambda_2 \) on the diagonal. The solution \( \mathbf{x}(t) \) is then obtained using the formula \( \mathbf{x}(t) = \mathbf{P}e^{\mathbf{Dt}}\mathbf{P}^{-1} \).
This solution represents the time evolution of the system, describing how the variables \( x_1 \) and \( x_2 \) change with respect to time \( t \) based on the matrix \( \mathbf{A} \). The eigenvalues and eigenvectors play a crucial role in understanding the behavior of the system over time.