Problem 3 - Calculus for Biology and Medicine

Question

Write each system of differential equations in matrix form. $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{3}-2 x_{1} \\ \frac{d x_{2}}{d t}=-x_{1} \end{array} $$

Step-by-Step Solution

Answer

Matrix form is: \(\frac{d}{dt}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -2 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\).

Step 1: Identify the Differential Equations

First, recognize the given system of differential equations: \(\frac{d x_{1}}{d t}=x_{3}-2 x_{1}\) and \(\frac{d x_{2}}{d t}=-x_{1}\). These are two equations with variables \(x_1\), \(x_2\), and \(x_3\).

Step 2: Express in Vector Form

Next, express the system of equations using vectors and matrices. Define a vector \(\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}\) and its derivative vector \(\frac{d\mathbf{x}}{dt} = \begin{bmatrix} \frac{dx_1}{dt} \ \frac{dx_2}{dt} \ \frac{dx_3}{dt} \end{bmatrix}\).

Step 3: Construct the Matrix Representation

Construct the matrix \(A\) such that the system \(\frac{d\mathbf{x}}{dt} = A\mathbf{x}\). Here, \(A\) must be a 3x3 matrix because the vector \(\mathbf{x}\) is 3x1. The elements of \(A\), must reflect the coefficients from the differential equations.

Step 4: Fill in the Matrix A

Look at the system of equations to fill the matrix:- From \(\frac{dx_1}{dt} = x_3 - 2x_1\), place \(-2\) at \(A[0,0]\), \(0\) at \(A[0,1]\), and \(1\) at \(A[0,2]\).- From \(\frac{dx_2}{dt} = -x_1\), place \(-1\) at \(A[1,0]\), \(0\) at \(A[1,1]\), and \(0\) at \(A[1,2]\).- As there is no \(\frac{dx_3}{dt}\) (\(dx_3/dt = 0\)), the third row will be all zeros.

Step 5: Write the Matrix Equation

Using the completed matrix \(A\), the system of differential equations can now be represented as: \[\frac{d}{dt} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} -2 & 0 & 1 \ -1 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}\]

Key Concepts

Matrix Representation

When dealing with systems of differential equations, it is often beneficial to express them in a matrix format. This method is commonly known as the matrix representation. By doing so, we simplify complex systems into a more manageable form that can be easily manipulated and solved using linear algebra techniques. For a given system, each differential equation helps to determine the corresponding matrix elements. The coefficients of each variable from the differential equations compose the elements of the matrix.
This process involves organizing these coefficients such that each equation corresponds to a row in the matrix, and each variable corresponds to a column. This transformation turns a set of possibly disparate equations into a unified matrix equation. Using the example provided, the system is converted into a matrix equation as follows:\[\frac{d}{dt} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} -2 & 0 & 1 \ -1 & 0 & 0 \ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}\]Here, the coefficients are carefully assigned based on the original system of equations. This matrix simplifies the manipulation and visualization of the underlying mathematical problem, offering a concise representation of the system dynamics.

Vector Form

Expressing differential equations in vector form is a key step in bridging the gap between intuitive understanding and mathematical analysis. It allows us to perform operations on these equations more efficiently. To achieve vector form representation, we first define a column vector comprising all the variables of the system. In our case, the vector is:\[\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}\]Each component of this vector represents a distinct variable from the system of equations. Next, we derive another vector that represents the derivatives of each respective component of \( \mathbf{x} \):\[\frac{d\mathbf{x}}{dt} = \begin{bmatrix} \frac{dx_1}{dt} \ \frac{dx_2}{dt} \ \frac{dx_3}{dt} \end{bmatrix}\]Using vector form, we capture the essence of the system dynamics in a streamlined manner. This approach benefits us by reducing not just the complexity but also the size of the system, thanks to the compact representation vectors provide. Ultimately, the goal is to write the equations as \( \frac{d\mathbf{x}}{dt} = A\mathbf{x} \), where \( A \) is a matrix mentioned in the previous section.

System of Equations

The concept of a system of equations is foundational when discussing differential equations in the context of matrices and vectors. A system of equations consists of multiple equations that are interrelated. Each equation comprises variables and consitutes part of a larger picture; they need to be considered as a whole.In our current example:- The system consists of two differential equations.

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

Each of these equations involves a derivative, illustrating how variable changes with respect to time. The interconnected nature of the system implies that the change in one variable affects others. By expressing these in terms of a standardized matrix equation, you align all parts of the system into a singular entity. This integrated approach is crucial for solving the equations, as it considers the relationship between variables rather than isolating them. Converting individual equations into a matrix makes applying advanced algorithms and computational methods feasible. The essence of formulating and solving systems of equations lies in understanding their matrix and vector representations, facilitating a more systematic approach to dissecting and resolving the relationships between multiple variables.