91Ó°ÊÓ

Differential equations are mathematical equations that relate a function with its derivatives. They allow us to describe various phenomena such as motion, heat, and waves. An ordinary differential equation involves functions of one variable and their derivatives. In our problem, we start with a system of two differential equations:

\(x^{\rprime} = -x + 3y, \quad y^{\rprime} = 3x + y\)

These equations describe how the functions \(x(t)\) and \(y(t)\) change over time. By observing their interaction through their derivatives, we can predict their behavior and evolution.

In a system of differential equations, the coefficients of each variable can be neatly organized into what we call a coefficient matrix. This matrix simplifies calculations and clearly displays the relationships between variables.

For our system, the coefficients are:

In the equation \(x^{\rprime} = -x + 3y\), the coefficients are -1 for \(x\) and 3 for \(y\).
In the equation \(y^{\rprime} = 3x + y\), the coefficients are 3 for \(x\) and 1 for \(y\).

Hence, the coefficient matrix is:
\(\begin{pmatrix} -1 & 3 \ 3 & 1 \end{pmatrix}\).

This matrix summarizes how each variable in the system contributes to the change in the others.

Vectors are essential tools in representing multiple quantities simultaneously. For our differential equations, we use vectors to group the functions and their derivatives.

The vector of functions for \(x\) and \(y\) is:

\(\begin{pmatrix} x \ y \end{pmatrix}\)

Similarly, the vector of derivatives is:

\(\begin{pmatrix} x^{\rprime} \ y^{\rprime} \end{pmatrix}\)

By using vectors, we can simplify the presentation and operations involving multiple quantities, making it easier to manage and manipulate them in calculations. It transforms our system of differential equations into a more compact and understandable form.

Combining our coefficient matrix with the vectors of functions and their derivatives, we can rewrite the system of differential equations as a single matrix equation. This form can be very powerful and easier to solve in many cases.

The combined equation looks like:

\(\begin{pmatrix} x^{\rprime} \ y^{\rprime} \end{pmatrix} = \begin{pmatrix} -1 \ 3 \ 3 \ 1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix}\)

This single matrix equation encapsulates all the information from the original system of differential equations. It shows how the derivatives of the variables are influenced by a linear combination of the variables themselves. This form is helpful for performing further mathematical operations and analyses, such as finding solutions or analyzing stability.