Problem 4 - Advanced Engineering Mathematics

Question

Write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=x-y \\ &\frac{d y}{d t}=x+2 z \\ &\frac{d z}{d t}=-x+z \end{aligned} $$

Step-by-Step Solution

Answer

\[ \frac{d}{dt}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 & -1 & 0 \\ 1 & 0 & 2 \\ -1 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} \]

Step 1: Identify the Variables and Equations

First, identify the variables and separate equations in the given system. Here, the variables are \(x\), \(y\), and \(z\). The given equations are: \(\frac{dx}{dt} = x - y\), \(\frac{dy}{dt} = x + 2z\), \(\frac{dz}{dt} = -x + z\).

Step 2: Express the System in Matrix Form

Express the system of equations in the form: \( \frac{d\vec{X}}{dt} = A\vec{X} \), where \(\vec{X}\) is the vector of variables and \(A\) is the coefficient matrix. Here, \(\vec{X} = \begin{bmatrix} x \ y \ z \end{bmatrix}\) and the equations can be rewritten as: \[\begin{align*}\frac{dx}{dt} &= 1\cdot x - 1\cdot y + 0\cdot z \frac{dy}{dt} &= 1\cdot x + 0\cdot y + 2\cdot z \frac{dz}{dt} &= -1\cdot x + 0\cdot y + 1\cdot z\end{align*}\]

Step 3: Write the Coefficient Matrix

Construct the coefficient matrix \(A\) from the equations: \[A = \begin{bmatrix} 1 & -1 & 0 \ 1 & 0 & 2 \ -1 & 0 & 1 \end{bmatrix}\]

Step 4: Write the System in Matrix Notation

Finally, combine \(\vec{X}\) and \(A\) to write the system in matrix form: \[\frac{d}{dt}\begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 1 & -1 & 0 \ 1 & 0 & 2 \ -1 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix}\]

Key Concepts

Linear Differential Equations

Linear differential equations arise when dealing with functions and their rates of change. In many real-world problems, such as physics and engineering, these equations are critical for modeling systems that change over time.
Linear differential equations differ from nonlinear in that all terms are either constant or linear in the unknown function and its derivatives. To qualify as linear, each term in the differential equation must be either a function multiplied by the unknown function without powers greater than one or a derivative of the unknown function.
In our exercise, we have a system of linear differential equations involving three variables: \(x\), \(y\), and \(z\). Each equation describes the rate of change of these variables over time, expressed as derivatives with respect to \(t\). By understanding how these variables interact and modify each other, you can gain insights into the behavior of complex systems over time.

Matrix Notation

Matrix notation is a concise way of writing systems of equations, allowing for easier manipulation and solution.Using matrices, you can represent multiple equations as a single entity, which can be very powerful when dealing with multiple variables and linear equations.
In our exercise, the system of equations is rewritten using matrix notation. You introduce a vector, \(\vec{X}\), which contains all the variables, in this case, \(x\), \(y\), and \(z\). Then there's the coefficient matrix \(A\), containing the constants that multiply each variable.
This matrix notation transforms the original cumbersome equations into a streamlined form: \(\frac{d\vec{X}}{dt} = A\vec{X}\). This concise representation is not just neat; it also aids in analyzing and solving the system efficiently using matrix algebra techniques.

Coefficient Matrix

The coefficient matrix plays a crucial role in transforming a system of linear equations into matrix form. It is essentially a layout of coefficients that influence each variable in the equations.
In the given exercise, the coefficient matrix \(A\) is extracted directly from the equations. It is a 3x3 matrix in this case because there are three equations. Here's how it looks:

• Row 1: Corresponds to the equation \(\frac{dx}{dt} = x - y\). Thus, the coefficients are 1 for \(x\), -1 for \(y\), and 0 for \(z\).
• Row 2: Corresponds to \(\frac{dy}{dt} = x + 2z\). Here, the coefficients are 1 for \(x\), 0 for \(y\), and 2 for \(z\).
• Row 3: For \(\frac{dz}{dt} = -x + z\), the coefficients are -1 for \(x\), 0 for \(y\), and 1 for \(z\).

This matrix, \(A\), captures the interdependencies between the variables and is instrumental in defining the dynamics of the system. It allows you to analyze and solve the system efficiently, making it easier to understand how changing one variable affects others.