91Ó°ÊÓ

To solve a system of differential equations, the first step is to write it in matrix form. This makes it easier to apply linear algebra techniques. For the given system of equations:
\( x^{\text{prime}} = -0.5x + y \) and \( y^{\text{prime}} = 0.5x \), we can represent them using a matrix.

Let’s denote our variables as a vector: \[ \begin{pmatrix} x \ y \end{pmatrix} \]
The system can be rewritten in the form:
\[ \frac{d}{dt} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} -0.5 & 1 \ 0.5 & 0 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} \]
This matrix form representation simplifies working with the equations and sets up the next steps in solving the system.

Eigenvalues play a crucial role in solving systems of differential equations. They help determine the behavior of the system over time. To find the eigenvalues of a matrix, we solve its characteristic equation.

For our matrix: \[ \begin{pmatrix} -0.5 & 1 \ 0.5 & 0 \end{pmatrix} \]
we start by finding the characteristic equation using the formula:
\[ \text{det}(A - \text{λ}I) = 0 \]
Here, \( A \) is our system matrix and \( \text{λ} \) represents an eigenvalue. Compute the determinant:
\[ \text{det} \begin{pmatrix} -0.5 - \text{λ} & 1 \ 0.5 & -\text{λ} \end{pmatrix} = 0 \]
Solving this will give us the eigenvalues. The next steps involve these eigenvalues to solve the system further.

The characteristic equation is derived by setting up the determinant of the matrix \( A - \text{λ}I \). This equation is key to finding the eigenvalues.

For our matrix, we set up:
\[ \text{det} \begin{pmatrix} -0.5 - \text{λ} & 1 \ 0.5 & -\text{λ} \end{pmatrix} = 0 \]
Calculate the determinant by multiplying the diagonal elements and subtracting the product of the off-diagonal elements:
\[ (-0.5 - \text{λ})(-\text{λ}) - (1 \times 0.5) = 0 \]
Expanding this would give a quadratic equation in \( \text{λ} \). Solving this quadratic equation will provide the eigenvalues, which indicate important properties of our system and help with finding the solution.

Once the eigenvalues are found, we can use them to find the general solution of the system. The eigenvalues help determine the eigenvectors. Combine the eigenvalues and eigenvectors to form the general solution.

From the eigenvalues, \( \text{λ}_1 \) and \( \text{λ}_2 \), assume solutions of the form:
\[ \begin{pmatrix} x \ y \end{pmatrix} = \textbf{v} e^{\text{λ}t} \]
where \( \textbf{v} \) is the eigenvector corresponding to eigenvalue \( \text{λ} \).
The general solution is a linear combination of these solutions:
\[ \textbf{X}(t) = C_1 \textbf{v}_1 e^{\text{λ}_1 t} + C_2 \textbf{v}_2 e^{\text{λ}_2 t} \]
Here, \( C_1 \) and \( C_2 \) are constants determined by initial conditions. This forms the complete solution to our system of differential equations.