Problem 8 - Biocalculus Calculus for the Life Sciences

Question

Write each system of linear differential equations in matrix notation. \(d x / d t=x-2, \quad d y / d t=2 y+3 x-1\)

Step-by-Step Solution

Answer

The matrix notation of the system is \( \mathbf{X}' = \begin{bmatrix} 1 & 0 \\ 3 & 2 \end{bmatrix} \mathbf{X} + \begin{bmatrix} -2 \\ -1 \end{bmatrix} \)."

Step 1: Identifying Variables and Derivatives

The system of linear differential equations is given as two separate equations: \( \frac{dx}{dt} = x - 2 \) and \( \frac{dy}{dt} = 2y + 3x - 1 \). Our task is to rewrite these equations in a matrix form.

Step 2: Formulating the Matrix Equation

First, we recognize that if we let \( \mathbf{X} = \begin{bmatrix} x \ y \end{bmatrix} \), the derivatives are \( \mathbf{X}' = \begin{bmatrix} \frac{dx}{dt} \ \frac{dy}{dt} \end{bmatrix} \). The differential equations can then be expressed in matrix form as \( \mathbf{X}' = A \mathbf{X} + \mathbf{B} \), where \( A \) is a matrix of coefficients for the variables, and \( \mathbf{B} \) is a vector representing constant terms in the equations.

Step 3: Determining Matrix A

Examine the coefficients of \( x \) and \( y \) in each differential equation. In \( \frac{dx}{dt} = x - 2 \), the coefficient of \( x \) is 1, and \( y \) is not present, so the coefficient is 0. Similarly, in \( \frac{dy}{dt} = 2y + 3x - 1 \), the coefficient of \( x \) is 3, and for \( y \) it is 2. Thus, matrix \( A \) can be written as \( \begin{bmatrix} 1 & 0 \ 3 & 2 \end{bmatrix} \).

Step 4: Identifying Constant Vector B

Since the terms \(-2\) and \(-1\) in the equations are constants and do not depend on \( x \) or \( y \), they form the vector \( \mathbf{B} = \begin{bmatrix} -2 \ -1 \end{bmatrix} \).

Step 5: Writing the Complete Matrix Form

Now, we can write the system in matrix form: \( \mathbf{X}' = A \mathbf{X} + \mathbf{B} \), which becomes \( \begin{bmatrix} \frac{dx}{dt} \ \frac{dy}{dt} \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 3 & 2 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} + \begin{bmatrix} -2 \ -1 \end{bmatrix} \).

Key Concepts

Matrix Notation

Matrix notation is a compact and efficient way to represent systems of equations, especially when dealing with linear differential equations. Instead of writing each equation separately, we combine them into a single matrix equation. This not only saves space but also makes the problem simpler to manipulate and solve.

In matrix notation, variables and their derivatives are bundled into vectors. For our original exercise, the variables are combined into a vector \( \mathbf{X} = \begin{bmatrix} x \ y \end{bmatrix} \). Their derivatives form another vector \( \mathbf{X}' = \begin{bmatrix} \frac{dx}{dt} \ \frac{dy}{dt} \end{bmatrix} \).

To express these in matrix form, we also introduce a coefficient matrix \( A \) and a constant vector \( \mathbf{B} \). This approach streamlines solving and analyzing the system, making operations like addition, multiplication, or even using computational tools significantly easier.

System of Differential Equations

A system of differential equations involves two or more equations that contain derivatives of multiple dependent variables with respect to one or more independent variables. In our example, we have a system consisting of two equations, each linking the derivatives of two variables, \(x\) and \(y\), to a combination of these variables.

Systems of differential equations are valuable because they allow us to model complex phenomena in areas like physics, biology, and economics where multiple changing quantities interact. For instance, this system could represent two interacting populations in an ecosystem or the interplay between supply and demand in a market.

By organizing these equations into a system, we can study their solutions collectively and understand the behavior of the variables over time. Solutions to systems often require numerical methods or analytical techniques tailored to the nature of the equations involved.

Matrix Form

The matrix form of a system of differential equations is a convenient representation that combines all equations into a single expression. This matrix equation is typically written as \( \mathbf{X}' = A\mathbf{X} + \mathbf{B} \). Here, \( \mathbf{X} \) is the vector of unknown functions, \( A \) is the matrix of coefficients, and \( \mathbf{B} \) is the vector of constant terms.

For the given problem, our matrix form becomes:

• \( A = \begin{bmatrix} 1 & 0 \ 3 & 2 \end{bmatrix} \)
• \( \mathbf{B} = \begin{bmatrix} -2 \ -1 \end{bmatrix} \)

This form helps to easily identify the structure of the system and the relationships between different parts. It simplifies solving the system either analytically or using computational algorithms that leverage matrix operations.

A key advantage of using the matrix form is the ability to apply sophisticated matrix techniques and mathematical tools, like eigenvalue analysis or Laplace transforms, to find and analyze solutions.

Matrix Vector Multiplication

Matrix vector multiplication is a fundamental operation in linear algebra, crucial for representing and solving systems of linear equations in matrix form. When you multiply a matrix \( A \) by a vector \( \mathbf{X} \), you perform dot products between the rows of \( A \) and the columns of \( \mathbf{X} \).

In our context, multiplying the matrix \( A = \begin{bmatrix} 1 & 0 \ 3 & 2 \end{bmatrix} \) by the vector \( \mathbf{X} = \begin{bmatrix} x \ y \end{bmatrix} \) results in:

• First element: \( 1 \times x + 0 \times y = x \)
• Second element: \( 3 \times x + 2 \times y = 3x + 2y \)

The resulting vector represents the interaction between the variables \( x \) and \( y \), as dictated by the coefficients of the matrix \( A \).

Understanding matrix vector multiplication allows us to interpret the effects of each variable on the system as a whole. It provides a precise mathematical framework for predicting the behavior of dynamic systems described by differential equations.