Problem 13 - Biocalculus Calculus for the Life Sciences

Question

Write each system of linear differential equations in matrix notation. \(d x / d t=x+4 y-3 t, \quad d y / d t=y-x\)

Step-by-Step Solution

Answer

The system is: \(\frac{d}{dt}\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 & 4 \\ -1 & 1 \end{bmatrix}\begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} -3t \\ 0 \end{bmatrix}\).

Step 1: Identify Variables and Equations

First, identify the variables and differential equations given. We have two variables, \(x\) and \(y\), and two differential equations: \(\frac{dx}{dt} = x + 4y - 3t\) and \(\frac{dy}{dt} = y - x\).

Step 2: Express Differential Equations in Vector Form

Write the differential equations in a vector form. Let \(\mathbf{X} = \begin{bmatrix} x \ y \end{bmatrix}\), then \(\frac{d\mathbf{X}}{dt} = \begin{bmatrix} \frac{dx}{dt} \ \frac{dy}{dt} \end{bmatrix}\).

Step 3: Identify the Coefficient Matrix

Identify the coefficient matrix by rewriting the differential equations focusing only on terms involving \(x\) and \(y\). The matrix is \(\mathbf{A} = \begin{bmatrix} 1 & 4 \ -1 & 1 \end{bmatrix}\), where diagonal elements come from self-variable multipliers (1 for \(x\) and \(y\)) and off-diagonals come from cross-variable multipliers (for example, 4 is the multiplier of \(y\) in the first equation).

Step 4: Write System in Matrix Notation

Combine the vector form and coefficient matrix to express the system: \(\frac{d\mathbf{X}}{dt} = \mathbf{A} \mathbf{X} + \mathbf{B}\). Here, \(\mathbf{B} = \begin{bmatrix} -3t \ 0 \end{bmatrix}\) represents the non-homogeneous part. So, the system in matrix form is:\[\frac{d}{dt}\begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 1 & 4 \ -1 & 1 \end{bmatrix}\begin{bmatrix} x \ y \end{bmatrix} + \begin{bmatrix} -3t \ 0 \end{bmatrix}.\]

Key Concepts

Matrix Notation

Matrix notation is a compact and organized way to represent systems of linear equations, including differential equations. By using matrices, we can group multiple equations into a single formula, making calculations more straightforward. This involves creating a matrix for the coefficients and vectors for variables and functions. Working with matrix notation can greatly simplify the process of solving differential equations, especially when dealing with systems that have multiple variables. Each equation in the system corresponds to a row in the matrix, and the coefficients of the variables form the columns.
Matrix notation is beneficial for visualizing the relationship among equations and for performing operations such as addition or multiplication on the system.

Vector Form

Expressing differential equations in vector form is an essential step in simplifying and solving complex systems. A vector groups related variables or functions into a single entity, which can then be manipulated collectively. For instance, in our exercise, we have two variables, \( x \) and \( y \). We can represent these using a single vector \( \mathbf{X} = \begin{bmatrix} x \ y \end{bmatrix} \).
This vector formation enables us to deal with systems of equations using matrix operations, which provide a clearer and more organized approach than handling each variable separately.Vectors help in unifying various parts of an equation, thereby simplifying the task of finding solutions or performing computations. This approach is also crucial for understanding and implementing various computational techniques in numerical analysis.

Coefficient Matrix

The coefficient matrix is a critical component in representing systems of linear equations using matrices. This matrix contains all the coefficients of the variables from each equation, arranged in a way that allows easy manipulation and calculation.In our exercise, the coefficient matrix \( \mathbf{A} \) is constructed from the terms involving \( x \) and \( y \) in the differential equations. It looks like this:

• Top row: coefficients for the first equation, leading to [1, 4]
• Bottom row: coefficients for the second equation, leading to [-1, 1]

The coefficient matrix helps to define the structure of the linear system and is integral in performing operations like finding solutions or determining stability. It provides a straightforward way to apply standard matrix techniques to the process of solving differential equations.

Non-Homogeneous Systems

Non-homogeneous systems of differential equations have terms that are not solely dependent on the variables and their derivatives. Such terms usually involve constants or functions of time, as seen in our exercise. In matrix terms, we represent these external influences using an additional vector, \( \mathbf{B} \), which accounts for all parts of the equations that do not involve the dependent variables directly. In our example, \( \mathbf{B} = \begin{bmatrix} -3t \ 0 \end{bmatrix} \) includes the term \( -3t \) from the first equation.
Dealing with non-homogeneous systems involves special techniques. These often include converting the system into a homogeneous one or finding a particular solution that addresses the additional terms. Understanding the non-homogeneous parts is crucial for applying the right method to find complete and accurate solutions.