Problem 1 - Calculus for Biology and Medicine

Question

In Problems \(1-4\), write each system of differential equations in matrix form. \(\begin{aligned} & \frac{d x_{1}}{d t}=2 x_{1}+3 x_{2} \\ \frac{d x_{2}}{d t} &=-x_{1}+x_{2} \end{aligned}\)

Step-by-Step Solution

Answer

The matrix form is \(\frac{d}{dt} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\).

Step 1: Understand the System of Equations

The given system of differential equations involves two equations with two variables \(x_1\) and \(x_2\) where \(\frac{d x_1}{d t}\) and \(\frac{d x_2}{d t}\) are expressed as linear combinations of \(x_1\) and \(x_2\).

Step 2: Express in Vector Notation

We represent the system of equations using vector notation. Let \(\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}\), and \(\frac{d\mathbf{x}}{dt} = \begin{bmatrix} \frac{d x_1}{d t} \ \frac{d x_2}{d t} \end{bmatrix}\). The system becomes: \[ \frac{d\mathbf{x}}{dt} = \begin{bmatrix} 2x_1 + 3x_2 \ -x_1 + x_2 \end{bmatrix} \]

Step 3: Identify the Coefficient Matrix

Identify the coefficients of \(x_1\) and \(x_2\) from each equation to form a matrix. The coefficients from the first equation are 2 and 3, and from the second equation are -1 and 1. This gives us the matrix: \[ A = \begin{bmatrix} 2 & 3 \ -1 & 1 \end{bmatrix} \]

Step 4: Write the System in Matrix Form

Using vector and matrix notation, express the original differential equations in matrix form as: \[ \frac{d\mathbf{x}}{dt} = A\mathbf{x} \] Substituting the identified matrix \(A\) and vector \(\mathbf{x}\), we have: \[ \frac{d}{dt} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 2 & 3 \ -1 & 1 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \]

Key Concepts

Understanding a System of Differential Equations

In the world of mathematics, particularly when dealing with calculus and differential equations, a **system of differential equations** comes into play when you have more than one differential equation working together. In this context, we are looking at a set of equations that explain how certain variables change over time.
Consider our provided system:

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

Here, both equations involve the variables \( x_1 \) and \( x_2 \). Each variable's rate of change depends not only on itself but also on the other variable. The goal is to solve this system by finding the functions \( x_1(t) \) and \( x_2(t) \). To analyze such systems efficiently, we use mathematical tools like vector notation and matrices.

The Role of Vector Notation in Systems of Differential Equations

Vector notation simplifies expressing multiple equations and unknowns. Instead of writing each equation separately, we bundle related equations into a structured form.
In our system:

• Variables: \( x_1 \) and \( x_2 \)
• Rates of change: \( \frac{d x_1}{d t} \) and \( \frac{d x_2}{d t} \)

This can be organized into vector form: \[\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \quad \text{and} \quad \frac{d\mathbf{x}}{dt} = \begin{bmatrix} \frac{d x_1}{d t} \ \frac{d x_2}{d t} \end{bmatrix}\]Vector notation allows us to express the entire system compactly and elegantly as \( \frac{d\mathbf{x}}{dt} = \begin{bmatrix} 2x_1 + 3x_2 \ -x_1 + x_2 \end{bmatrix} \). Such concise representation paves the way for using matrices, aiding in the resolution of systems with even more variables.

Understanding and Utilizing a Coefficient Matrix

The **coefficient matrix** is a cornerstone of linear algebra, used to handle systems of equations, especially differential systems, more effectively. It captures the coefficients directly affecting each variable in the system.
From our given system:

• The equation \( \frac{d x_{1}}{d t} = 2x_1 + 3x_2 \) contributes coefficients 2 and 3.
• The equation \( \frac{d x_{2}}{d t} = -x_1 + x_2 \) gives coefficients -1 and 1.

Arranging these, the coefficient matrix \( A \) is:\[A = \begin{bmatrix} 2 & 3 \ -1 & 1 \end{bmatrix}\]The matrix form of our system is hence demonstrated as:\[\frac{d\mathbf{x}}{dt} = A\mathbf{x}\]Such a form leverages the power of linear algebra, making it easier to analyze and solve systems, especially when working with larger, complex systems of differential equations. Understanding the role and construction of the coefficient matrix is key to mastering such mathematical problems.