91Ó°ÊÓ

Non-homogeneous differential equations have an extra term in the equation. This term is called the "non-homogeneous" part and differentiates these from the simpler, homogeneous equations.
In the given exercise, the non-homogeneous system of equations is:

• \( \frac{d x}{d t} = 5x + 9y + 2 \)
• \( \frac{d y}{d t} = -x + 11y + 6 \)

These additions of \( +2 \) and \( +6 \) in the equations are what make this system non-homogeneous.
To solve such systems, we develop the complete solution in two parts:

• First, the homogeneous solution that involves differential terms without the constants.
• Second, the particular solution that accounts for the non-homogeneous part.

By adding these solutions, we obtain the complete general solution for the system. The method of undetermined coefficients is a common technique used to find the particular solution, as illustrated in the exercise.
It involves proposing a form for the solution and determining the unknown coefficients by substituting back into the original equations.

Eigenvalues and eigenvectors play a crucial role in finding solutions to differential equations. In the given system, we transform it into matrix form:

• \( \frac{d}{dt} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 5 & 9 \ -1 & 11 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \)

For the homogeneous part, we need to solve for eigenvalues and eigenvectors of the matrix \(A\).
The eigenvalues are found by determining the roots of the characteristic equation:

• \(|A - \lambda I| = 0\)
• \( \lambda^2 - 16\lambda + 56 = 0 \)

Solving this quadratic equation yields \( \lambda_1 = 8 \) and \( \lambda_2 = 8 \), indicating a repeated eigenvalue.
With these eigenvalues, we find the associated eigenvectors which are crucial for constructing the solution.
The eigenvector for \( \lambda_1 = 8 \) can be computed by solving:

• \( \begin{bmatrix} -3 & 9 \ -1 & 3 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix} \)

This gives the eigenvector \( \begin{bmatrix} 3 \ 1 \end{bmatrix} \). Eigenvalues and eigenvectors together allow us to construct the homogeneous solution essential to solving the differential system.

Homogeneous solutions form the first part of the complete solution for non-homogeneous differential systems. For the system given in the exercise, the homogeneous equation is:

• \( \frac{d}{dt} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 5 & 9 \ -1 & 11 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \)

To find the homogeneous solution, we use the eigenvalues and eigenvectors derived from the matrix \(A\).
The homogeneous solution is based on the general structure using these eigenvalues. In this case, since the eigenvalue \( \lambda = 8 \) is repeated, the solution takes the form:

• \( \mathbf{x}_h(t) = C_1 e^{8t} \begin{bmatrix} 3 \ 1 \end{bmatrix} + C_2 t e^{8t} \begin{bmatrix} 3 \ 1 \end{bmatrix} \)

Here, \( C_1 \) and \( C_2 \) are arbitrary constants determined by initial conditions. This solution accounts for the response of the system to the structure defined by \(A\) without considering the non-homogeneous terms.
It forms the foundational piece upon which we build the complete solution by adding the particular solution.