91Ó°ÊÓ

A homogeneous system of differential equations is one where there is no external input (like \( \cos 2t \) in our exercise), meaning that any solution can be scaled by a constant to produce another solution. Understanding this is crucial because the solutions of the homogeneous system form the complementary part of our final solution when dealing with non-homogeneous systems.

In the given exercise, the homogeneous system is derived by setting the non-homogeneous part to zero: \[ \begin{bmatrix} x_h' \ y_h' \end{bmatrix} = \begin{bmatrix} 1 & -5 \ 1 & -1 \end{bmatrix} \begin{bmatrix} x_h \ y_h \end{bmatrix}\] This simplification allows us to focus on the intrinsic properties of the system without external forcing. The natural behavior of the system as time progresses is characterized by the solutions of this homogeneous equation.

These natural behaviors are described by exponential functions that are determined by the system's eigenvalues. They provide a base solution set to which we will add particular solutions that match our specific case.

Eigenvalues and eigenvectors play a pivotal role in solving systems of linear equations, including differential equations in matrix form.

To find the eigenvalues, we form the characteristic equation of the system matrix by subtracting \( \lambda \) times the identity matrix from the coefficient matrix and then setting the determinant to zero. In our exercise, this results in: \[ \lambda^2 - \lambda - 6 = 0 \] Factoring this gives eigenvalues \( \lambda_1 = 3 \) and \( \lambda_2 = -2 \).

With these eigenvalues, we next determine the eigenvectors by substituting back into the modified system. For each eigenvalue, solving: \[ \begin{bmatrix} 1 - \lambda & -5 \ 1 & -1 - \lambda \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \]

The eigenvectors correspond to these solutions and they describe the direction in which the system evolves. They are essential for constructing the homogeneous solution, which forms the core of the overall solution to our differential system.

A particular solution of a differential equation solves the non-homogeneous part, accounting specifically for external inputs like \( \cos 2t \) in our exercise.

We guess a solution structure based on the form of the non-homogeneous term (e.g., cosine terms imply trying sine and cosine terms). Thus, we assume a particular solution of the form: \[ \begin{bmatrix} x_p \ y_p \end{bmatrix} = \begin{bmatrix} A \cos 2t + B \sin 2t \ C \cos 2t + D \sin 2t \end{bmatrix} \]

By substituting this assumed solution into the original non-homogeneous system of equations and differentiating, we substitute back and solve for unknown coefficients \( A, B, C, \) and \( D \) by equating terms of like functions (harmonic functions in this case). These values give us the specific response of the system to the external forcing term.

Differential equations in matrix form allow for a more structured approach to finding solutions for systems of equations. This representation inevitably leads us to explore concepts like eigenvalues and eigenvectors more prominently.

For our exercise, we rewrite the coupled differential equations using matrices to systematically address the problem: \[ \begin{bmatrix} x' \ y' \end{bmatrix} = \begin{bmatrix} 1 & -5 \ 1 & -1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} + \begin{bmatrix} \cos 2t \ 0 \end{bmatrix} \]

This setup emphasizes the relationship between different components of the system through linear transformations. The transformation matrix contains all the information about the system's natural changes, while the additional vector represents external inputs or forcing functions, guiding us to find the particular solution.

Using matrix forms streamlines the calculation and visualization of solutions, making it easier to solve even complex systems efficiently.