Problem 4 - Calculus for Biology and Medicine

Question

Write each system of differential equations in matrix form. $$ \begin{array}{l} \frac{d x_{1}}{d t}=2 x_{2}-3 x_{1}-x_{3} \\ \frac{d x_{2}}{d t}=-x_{1}+x_{2} \end{array} $$

Step-by-Step Solution

Answer

The system in matrix form is \( \frac{d\mathbf{X}}{dt} = \begin{bmatrix} -3 & 2 & -1 \\ -1 & 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \).

Step 1: Understand the System of Differential Equations

We have two differential equations with three variables: 1. \( \frac{d x_{1}}{d t}=2 x_{2}-3 x_{1}-x_{3} \) 2. \( \frac{d x_{2}}{d t}=-x_{1}+x_{2} \). These equations describe how \( x_1 \) and \( x_2 \) change with respect to time \( t \).

Step 2: Identify Coefficients for Matrix Representation

Identify the coefficients of each variable from both equations. For the first equation \( \frac{d x_{1}}{d t}=2 x_{2}-3 x_{1}-x_{3} \), the coefficients are: - For \( x_1 \): -3 - For \( x_2 \): 2 - For \( x_3 \): -1. For the second equation \( \frac{d x_{2}}{d t}=-x_{1}+x_{2} \), the coefficients are: - For \( x_1 \): -1 - For \( x_2 \): 1 - For \( x_3 \): 0.

Step 3: Construct the Coefficient Matrix

Based on the coefficients identified, form a matrix. The coefficients for the variables \( x_1, x_2, \) and \( x_3 \) for the two equations form the matrix: \[ A = \begin{bmatrix} -3 & 2 & -1 \ -1 & 1 & 0 \end{bmatrix} \] This is called the coefficient matrix \( A \).

Step 4: Write the System in Matrix Form

Express the system of differential equations in matrix form using the coefficient matrix \( A \), the vector of derivatives, and the vector of variables: Let \( \mathbf{X} = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} \) and \( \frac{d\mathbf{X}}{dt} = \begin{bmatrix} \frac{dx_1}{dt} \ \frac{dx_2}{dt} \ \frac{dx_3}{dt} \end{bmatrix} \). The system in matrix form is: \[ \frac{d\mathbf{X}}{dt} = A \cdot \mathbf{X} = \begin{bmatrix} -3 & 2 & -1 \ -1 & 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} \]

Key Concepts

Coefficient Matrix

The coefficient matrix is a crucial element when transforming a system of differential equations into matrix form. It is a compact way of organizing the coefficients from each equation into a structured grid.
This matrix represents the relationship between variables in a system of equations. For instance, in our specific system:

• The first differential equation \( \frac{d x_{1}}{d t}=2 x_{2}-3 x_{1}-x_{3} \) gives us the coefficients: \(-3\) for \(x_1\), \(2\) for \(x_2\), and \(-1\) for \(x_3\).
• The second differential equation \( \frac{d x_{2}}{d t}=-x_{1}+x_{2} \) provides: \(-1\) for \(x_1\), \(1\) for \(x_2\), and \(0\) for \(x_3\). This zero indicates the absence of \(x_3\).

Combining these coefficients forms the coefficient matrix \(A\) that looks like:\[A = \begin{bmatrix} -3 & 2 & -1 \ -1 & 1 & 0 \end{bmatrix}\]The rows correspond to the respective differential equations, and the columns correspond to the position of each variable. This matrix provides an easy visual to assess influences each variable has on another over time.

Differential Equations

Differential equations are equations that involve an unknown function and its derivatives. They describe how a specific function changes over time or space. These equations can be quite complex, but they are fundamental in modeling real-world scenarios such as motion, heat, population dynamics, and more.
In the given exercise, we consider two differential equations:

• \( \frac{d x_{1}}{d t} = 2x_2 - 3x_1 - x_3 \)
• \( \frac{d x_{2}}{d t} = -x_1 + x_2 \)

These define how each variable, \(x_1\) and \(x_2\), changes with respect to time, as they are dependent on themselves and other variables like \(x_3\). The derivatives \(\frac{d x_{1}}{d t}\) and \(\frac{d x_{2}}{d t}\) depict the rate of change for the variables. Understanding how to read and set up these equations is necessary to translate them into a matrix form, which simplifies their analysis.

System of Equations

A system of equations consists of multiple, interrelated equations involving several variables to solve together. When working with systems of differential equations, they showcase how variables interact in tandem, affecting each other's rates of change.
In our exercise, the system includes:

• \( \frac{d x_{1}}{d t} = 2 x_2 - 3 x_1 - x_3 \)
• \( \frac{d x_{2}}{d t} = -x_1 + x_2 \)

These equations can initially appear separate, but they often hinge on common variables linking them. By expressing this system in matrix form, we use a simplified, systematic way to handle it mathematically. Matrix equations reveal interactions via a matrix-vector multiplication, as seen:\[\frac{d\mathbf{X}}{dt} = A \cdot \mathbf{X}\]where \(\mathbf{X} = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}\). This allows for straightforward computational methods to solve complex systems, making it a valuable tool for engineers, scientists, and mathematicians.