Problem 3 - Calculus for Biology and Medicine

Question

Write each system of differential equations in matrix form. \(\frac{d x_{1}}{d t}=x_{3}-2 x_{1}\) \(\frac{d x_{2}}{d t}=-x_{1}+x_{3}\) \(\frac{d x_{3}}{d t}=x_{1}+x_{2}+x_{3}\)

Step-by-Step Solution

Answer

The system of equations in matrix form is \[\frac{d\mathbf{x}}{dt} = A\mathbf{x}\] where \(A = \begin{pmatrix}-2 & 0 & 1 \\ -1 & 0 & 1 \\ 1 & 1 & 1\end{pmatrix}\).

Step 1: Understand the System of Equations

We start by analyzing each given differential equation. We have three equations involving three variables \(x_1\), \(x_2\), and \(x_3\). The derivatives \(\frac{dx_1}{dt}\), \(\frac{dx_2}{dt}\), and \(\frac{dx_3}{dt}\) all depend on the variables \(x_1\), \(x_2\), and \(x_3\).

Step 2: Arrange Variables and Coefficients

Identify the coefficients of each variable in each equation. For \(\frac{dx_1}{dt}\), the coefficients are \(x_3\) and \(-2x_1\), which correspond to \([0, 0, 1]\) in the matrix. For \(\frac{dx_2}{dt}\), the coefficients are \(-x_1\) and \(x_3\), represented as \([-1, 0, 1]\). Lastly, for \(\frac{dx_3}{dt}\), the coefficients \( [1, 1, 1] \) correspond to the sum of all variables.

Step 3: Write the System in Matrix Form

We can represent the system of equations using matrices. The derivative vector is \(\begin{pmatrix}\frac{dx_1}{dt} \ \frac{dx_2}{dt} \ \frac{dx_3}{dt}\end{pmatrix}\), the variable vector is \(\begin{pmatrix}x_1 \ x_2 \ x_3\end{pmatrix}\), and the coefficient matrix is \(\begin{pmatrix}-2 & 0 & 1 \ -1 & 0 & 1 \ 1 & 1 & 1\end{pmatrix} \). Hence, the system in matrix form is: \[\begin{pmatrix}\frac{dx_1}{dt} \\frac{dx_2}{dt} \\frac{dx_3}{dt}\end{pmatrix}=\begin{pmatrix}-2 & 0 & 1 \-1 & 0 & 1 \1 & 1 & 1\end{pmatrix}\begin{pmatrix}x_1 \x_2 \x_3\end{pmatrix}\]

Key Concepts

Matrix Form

In mathematics, expressing systems of equations in matrix form is a powerful technique.
Basically, this involves organizing both variables and their coefficients into matrices to simplify complex calculations.
In our example problem, we have a system of differential equations. We'll convert this into a convenient matrix form.

Consider a matrix as an organized rectangular arrangement of numbers in rows and columns. For this problem, you'll see:

• The variable matrix to represent the variables, for instance, \( \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} \).
• The coefficient matrix hosts the numerical coefficients from each equation.
• The derivative matrix, which will be \( \begin{pmatrix} \frac{dx_1}{dt} \ \frac{dx_2}{dt} \ \frac{dx_3}{dt} \end{pmatrix} \), shows how each variable changes.

The matrix equation forms as:
\[ \begin{pmatrix} \frac{dx_1}{dt} \ \frac{dx_2}{dt} \ \frac{dx_3}{dt} \end{pmatrix} = \begin{pmatrix} -2 & 0 & 1 \ -1 & 0 & 1 \ 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} \]
In essence, adopting matrix form simplifies the expression of complex systems, allowing for more straightforward computation and analysis.

Systems of Differential Equations

The world of differential equations deals with equations involving derivatives, expressing rates of change.
Systems of differential equations occur when there are two or more interrelated differential equations.
These systems are incredibly useful for modeling dynamic processes in various fields such as engineering, physics, and biology.

Let's break it down using our example. In our previous problem, we have a system involving the derivatives of \( x_1, x_2, \text{ and } x_3 \):

• \( \frac{dx_1}{dt} = x_3 - 2x_1 \)
• \( \frac{dx_2}{dt} = -x_1 + x_3 \)
• \( \frac{dx_3}{dt} = x_1 + x_2 + x_3 \)

This system of differential equations shows that each of these three variables depend on each other in some manner.
Our task is to solve for each variable considering how they interdepend.
This often requires applying methods such as matrix algebra, as seen before, which helps convert these equations into a form easier to handle.

Coefficient Matrix

A critical element when dealing with equations in matrix form is the coefficient matrix.
This matrix contains all the coefficients from the system of equations.
The organization ensures each row corresponds to an equation, and each column corresponds to a variable.

In our example, the coefficient matrix itself looks like this: \[ \begin{pmatrix} -2 & 0 & 1 \ -1 & 0 & 1 \ 1 & 1 & 1 \end{pmatrix} \]
Each row here contains the coefficients from one of the equations:

• The first row (\(-2, 0, 1\)) comes from the equation \( \frac{dx_1}{dt} = x_3 - 2x_1 \).
• The second row (\(-1, 0, 1\)) corresponds to \( \frac{dx_2}{dt} = -x_1 + x_3 \).
• The third row (\(1, 1, 1\)) comes from \( \frac{dx_3}{dt} = x_1 + x_2 + x_3 \).

This matrix organizes and maintains the relationship between variables.
Understanding and constructing the coefficient matrix is crucial as it forms the backbone of solving systems of differential equations using matrix methods.