91Ó°ÊÓ

When facing a system of differential equations, one of the first steps to solving it involves finding the eigenvalues of the associated matrix. Eigenvalues are special scalars in a mathematical equation, and they reveal important properties about a system. In the context of differential equations, eigenvalues help determine the behavior and stability of each solution component.
For a given matrix, the eigenvalues are found by solving the characteristic equation, which is derived from the matrix minus a scaled identity matrix: \[ \det(A - \lambda I) = 0 \]
In our original problem, the matrix \( A = \begin{bmatrix} 1 & 1 \ 1 & -1 \end{bmatrix} \) leads us to solve the equation \( \lambda^2 - \lambda - 2 = 0 \), resulting in eigenvalues \( \lambda = 2 \) and \( \lambda = -1 \).
This means our system has two distinct eigenvalues, which play a critical role in constructing the general solution.

Eigenvectors are vectors that, when multiplied by the matrix, do not change direction, only their magnitude, which is scaled by the eigenvalues. Each eigenvalue has a corresponding eigenvector, and these eigenvectors form the basis of the solution space for the system of equations.
For eigenvalue \( \lambda = 2 \) in our problem, the associated eigenvector is found by solving: \((A - 2I)\mathbf{v} = 0\). This results in \( \begin{bmatrix} -1 & 1 \ 1 & -3 \end{bmatrix}\begin{bmatrix} v_1 \ v_2 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \), simplifying to \( v_1 = v_2 \). Thus, a valid eigenvector is \( \begin{bmatrix} 1 \ 1 \end{bmatrix} \).
Similarly, for \( \lambda = -1 \), the eigenvector \( \begin{bmatrix} 1 \ -2 \end{bmatrix} \) is found through \((A + I)\mathbf{v} = 0\), leading to the system \( 2v_1 + v_2 = 0 \).
These eigenvectors allow us to construct the general solution of the differential equation.

Initial conditions are specific values given for the solution at a particular time, usually \( t = 0 \). They help determine the exact trajectory a solution will take out of the infinite solutions in a differential system.
In our problem, initial conditions are \( x_1(0) = 1 \) and \( x_2(0) = 0 \).
To incorporate these into the general solution, we substitute these values to solve for the constants, \( c_1 \) and \( c_2 \), in the general equation: \[ x_1(t) = c_1 e^{2t} + c_2 e^{-t} \] \[ x_2(t) = c_1 e^{2t} - 2c_2 e^{-t} \]
By applying the initial conditions, we set up equations: \[ 1 = c_1 + c_2 \] \[ 0 = c_1 - 2c_2 \] leading us to the specific values \( c_1 = \frac{2}{3} \) and \( c_2 = \frac{1}{3} \). These constants precisely define the specific trajectory of the solution.

Converting a system of differential equations into matrix form is a crucial step that allows us to utilize linear algebra techniques to find a solution. By expressing the system in terms of matrices, we can work with eigenvalues and eigenvectors to determine the behavior of the system over time.
The original system of equations: \( x_1' = x_1 + x_2 \) and \( x_2' = x_1 - x_2 \) can be written in matrix form as: \[ \mathbf{x}' = A\mathbf{x} \] where \( A = \begin{bmatrix} 1 & 1 \ 1 & -1 \end{bmatrix} \) and \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \).
This transformed representation simplifies the system into a more manipulable form, allowing us to systematically approach finding the general and specific solutions. Using matrices also clarifies the impact of each component on the system dynamics, enhancing our understanding of how solutions evolve.