Problem 7 - Biocalculus Calculus for the Life Sciences

Question

Write each system of linear differential equations in matrix notation. \(d x / d t=5 x-3 y, \quad d y / d t=2 y-x\)

Step-by-Step Solution

Answer

The system in matrix notation is: \( \frac{d}{dt} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 & -3 \\ -1 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \).

Step 1: Identify the System

We are given a system of linear differential equations:1. \( \frac{dx}{dt} = 5x - 3y \)2. \( \frac{dy}{dt} = 2y - x \)Our task is to express these equations in matrix notation.

Step 2: Write the Vector Form

Introduce a vector \( \mathbf{X} \) such that \( \mathbf{X} = \begin{bmatrix} x \ y \end{bmatrix} \). We also express the derivatives as \( \frac{d\mathbf{X}}{dt} = \begin{bmatrix} \frac{dx}{dt} \ \frac{dy}{dt} \end{bmatrix} \).

Step 3: Construct the Coefficient Matrix

Identify the coefficients of \( x \) and \( y \) from the equations:- For \( \frac{dx}{dt} = 5x - 3y \), the coefficients are \( 5 \) and \( -3 \).- For \( \frac{dy}{dt} = 2y - x \), rewrite it as \( \frac{dy}{dt} = -x + 2y \), so the coefficients are \( -1 \) and \( 2 \).Thus, the coefficient matrix \( A \) is \( \begin{bmatrix} 5 & -3 \ -1 & 2 \end{bmatrix} \).

Step 4: Express as Matrix Equation

Combine the vector form and coefficient matrix to write the system as a matrix equation:\[ \frac{d\mathbf{X}}{dt} = A\mathbf{X} \]Where \( A = \begin{bmatrix} 5 & -3 \ -1 & 2 \end{bmatrix} \) and \( \mathbf{X} = \begin{bmatrix} x \ y \end{bmatrix} \).Therefore the matrix equation is:\[ \frac{d}{dt} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 5 & -3 \ -1 & 2 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \]

Step 5: Verify the Matrix Equation

Ensure that the matrix equation correctly represents the original system. Multiplied out, we have:\[ \begin{bmatrix} \frac{dx}{dt} \ \frac{dy}{dt} \end{bmatrix} = \begin{bmatrix} 5 & -3 \ -1 & 2 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \]This results in the original:1. \( \frac{dx}{dt} = 5x - 3y \)2. \( \frac{dy}{dt} = -x + 2y \)The matrix equation is verified.

Key Concepts

Linear Differential Equations

Linear differential equations are equations that involve unknown functions and their derivatives. These types of equations arise naturally in various scientific fields when modeling real-world phenomena. They are crucial in describing processes that change linearly over time, such as circuits and mechanical vibrations. For an equation to be classified as linear, it needs to meet certain criteria:

• The function and its derivatives are of the first degree (no squares, cubes, etc.).
• There are no products of the unknown function and its derivatives within the equation.
• All coefficients involved are functions of the independent variable, typically time, or constants.

We refer to these equations as 'linear' because if you picture them as graphs or geometric shapes, they form straight lines or planes in their simplest form.
Understanding their behavior is essential, as linear equations permit straightforward methods of solution and lend themselves well to analysis and comprehension.

Coefficient Matrix

The coefficient matrix is a pivotal concept when working with systems of differential equations in matrix notation. It serves as a compact representation of the coefficients of the variables from the differential equations. Constructing it involves identifying the coefficients from each equation and organizing them into an array format.
In our example, the system of equations:

• \( \frac{dx}{dt} = 5x - 3y \)
• \( \frac{dy}{dt} = -x + 2y \)

leads to the creation of the coefficient matrix.The matrix \( A \) is made by placing the coefficients into a matrix form:\[A = \begin{bmatrix} 5 & -3 \ -1 & 2 \end{bmatrix}\]This matrix becomes central in writing the differential equation system in a simplified, more manageable matrix notation which can then be analyzed further or used for computational solutions.

System of Differential Equations

A system of differential equations consists of multiple equations that share a common set of variables. These systems are used to model complex systems where two or more processes or states interact over time. They are vital in fields like physics, engineering, and biology.
In the context of our example, solving the system requires expressing the equations in a more unified form. By assembling the equations into a matrix format, the system \( \frac{d\mathbf{X}}{dt} = A\mathbf{X} \) emerges, allowing us to leverage powerful mathematical tools like eigenvector analysis and matrix exponentiation.
Key benefits of using such systems include:

• Simplification by combining equations to form compact expressions.
• Facilitation of analytical or numerical solutions.
• Enhanced ability to predict long-term behavior through characteristic methods.

Understanding and manipulating systems of differential equations offer the ability to control, alter, and understand the dynamics of many practical systems.