Problem 10 - Biocalculus Calculus for the Life Sciences

Question

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

Step-by-Step Solution

Answer

The system is \( \frac{d}{dt} \begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = \begin{bmatrix} 0 & 5 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} x(t) \\ y(t) \end{bmatrix} \).

Step 1: Identify the Variables

First, identify the variables in the system which are dependent on the independent variable. Here we have two functions: \(x(t)\) and \(y(t)\), both of which are dependent on \(t\).

Step 2: Understand the System of Equations

We have two equations: (1) \( \frac{d x}{d t} = 5y \) and (2) \( \frac{d y}{d t} = 2x - y \). These equations express the rates of change of the variables \(x\) and \(y\).

Step 3: Write the System in Matrix Form

In matrix form, a system of linear differential equations can be written as \( \frac{d}{dt} \begin{bmatrix} x(t) \ y(t) \end{bmatrix} = A \begin{bmatrix} x(t) \ y(t) \end{bmatrix} \), where \(A\) is the matrix of coefficients. Identify the coefficients from the given equations.

Step 4: Construct the Matrix of Coefficients

From the equations, construct the matrix of coefficients \(A\). For the first equation, the coefficient for \(y\) is 5 and for \(x\) is 0. For the second equation, the coefficients are 2 for \(x\) and -1 for \(y\). Thus, \(A = \begin{bmatrix} 0 & 5 \ 2 & -1 \end{bmatrix}\).

Step 5: Write the Final Matrix Form of the System

Now that we have identified \(A\), express the system as a matrix differential equation: \[ \frac{d}{dt} \begin{bmatrix} x(t) \ y(t) \end{bmatrix} = \begin{bmatrix} 0 & 5 \ 2 & -1 \end{bmatrix} \begin{bmatrix} x(t) \ y(t) \end{bmatrix} \].

Key Concepts

Linear Differential Equations

Linear differential equations involve derivatives of a function or functions with respect to one or more variables.They are called 'linear' because their structure is linear in terms of the unknown function and its derivatives.In simple terms, there are no products or powers of the variables involved in the equations.
For example, in our original system, we have the equations:

• \( \frac{d x}{d t} = 5y \)
• \( \frac{d y}{d t} = 2x - y \)

Each of these equations describes the rate at which one of the variables changes with respect to the independent variable, here being \(t\).
This is why such equations are useful in understanding processes where changes in some quantities depend linearly on changes in other quantities.

Matrix Form

The matrix form of a system of differential equations provides a compact representation.It combines all equations into a unified structure using matrices, making it easier to manipulate and solve them, especially when dealing with numerous variables.
In this exercise, we put two differential equations into matrix notation:

• Matrix equations are typically written as:\[\frac{d}{dt} \begin{bmatrix} x(t) \ y(t) \end{bmatrix} = A \begin{bmatrix} x(t) \ y(t) \end{bmatrix}\]

Here, the matrix \(A\) contains the coefficients from the differential equations.
Creating this matrix involves extracting coefficients from each equation, which in our problem resulted in: \[A = \begin{bmatrix} 0 & 5 \ 2 & -1 \end{bmatrix}\] This form simplifies many processes like finding solutions and understanding the behavior of the system.

System of Equations

A system of equations is a collection of two or more equations with a set of variables.In differential equations, this means we have multiple interconnected equations that describe how different variables change relative to each other with respect to an independent variable.
Our given system:

• \( \frac{d x}{d t} = 5y \)
• \( \frac{d y}{d t} = 2x - y \)

shows two relationships.Each equation gives us a clue about how the value of one variable relies on the other.
This interconnectedness makes the system complex yet allows us to capture a more significant understanding of the variables' dynamics.By expressing it in a matrix, it becomes easier to simultaneously address these relationships and find collective solutions.

Rate of Change

The rate of change in differential equations describes how quickly a variable changes over time or with respect to another variable. In our context, it's the derivative that shows how fast \(x(t)\) or \(y(t)\) is changing concerning time \(t\).

• The equation \( \frac{d x}{d t} = 5y \) describes how \(x\) changes at a rate proportional to the current state of \(y\).
• Similarly, \( \frac{d y}{d t} = 2x - y \) shows that the change in \(y\) is determined by both \(x\) and itself.

The coefficients in each expression hint at the influence one variable has on the rate of another. This is crucial, as differential equations provide insights into dynamic behavior, such as predicting future states or understanding oscillations and stability within systems.