Problem 1 - Calculus for Biology and Medicine

Question

Write each system of differential equations in matrix form. $$ \begin{array}{l} \frac{d x_{1}}{d t}=2 x_{1}+3 x_{2} \\ \frac{d x_{2}}{d t}=-4 x_{1}+x_{2} \end{array} $$

Step-by-Step Solution

Answer

The matrix form is \(\frac{d}{dt}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ -4 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\).

Step 1: Identify the Structure

The given system of differential equations is composed of linear combinations of the variables \(x_1\) and \(x_2\) on the right-hand sides of the equations. Our goal is to represent this system in matrix form.

Step 2: Write Equations in Standard Form

We have two equations: \(\frac{d x_{1}}{d t}=2 x_{1}+3 x_{2}\)\(\frac{d x_{2}}{d t}=-4 x_{1}+x_{2}\)This indicates that each derivative is a combination of the variables \(x_1\) and \(x_2\).

Step 3: Define the Variable Matrix

Define the variable vector \(\mathbf{x}\) as:\[\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}\]

Step 4: Define the System Matrix

Determine the matrix \(A\) of coefficients that multiplies the vector \(\mathbf{x}\). From the equations:\[A = \begin{bmatrix} 2 & 3 \ -4 & 1 \end{bmatrix}\]

Step 5: Formulate the Matrix Equation

Use the definitions from the previous steps to express the differential system in matrix form:\[\frac{d\mathbf{x}}{dt} = A \mathbf{x}\]This results in the matrix equation:\[\frac{d}{dt}\begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 2 & 3 \ -4 & 1 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix}\]

Step 6: Summary

Thus, the given system of differential equations is written in matrix form as \(\frac{d\mathbf{x}}{dt} = A \mathbf{x}\), where \(A = \begin{bmatrix} 2 & 3 \ -4 & 1 \end{bmatrix}\).

Key Concepts

Matrix Form

Matrix form provides a compact way to represent systems of equations, especially when dealing with multiple variables and constants. Instead of writing out numerous equations, you use matrices and vectors to simplify the representation.
Here’s how it works:

• First, identify the variables involved. In our exercise, we have variables \(x_1\) and \(x_2\).
• Next, collect these variables into a vector, \( \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \end{bmatrix} \).
• Determine the coefficients of the variables from each equation to form the matrix, \(A\). For example, in the equation \( \frac{d x_1}{d t} = 2x_1 + 3x_2 \), the coefficients are 2 and 3.

Using these steps, the system of differential equations can be neatly expressed in the format \( \frac{d\mathbf{x}}{dt} = A\mathbf{x} \). This method not only streamlines mathematical computations but also makes it easier to apply linear algebra techniques.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear transformations. It provides essential tools for dealing with systems of linear equations and is foundational for understanding differential equations in matrix form.
Key concepts in linear algebra include:

• Vectors: These are ordered lists of numbers, and in our context, they represent the state variables, \(x_1\) and \(x_2\).
• Matrices: These are rectangular arrays of numbers that can represent a linear transformation or system of equations like matrix \(A\).
• Matrix Multiplication: This operation combines two matrices or a matrix and a vector to produce another matrix or vector, useful in computing \( A\mathbf{x} \).
• Determinants and Eigenvalues: These measures help assess the properties of matrices and how they transform vectors.

Understanding these concepts allows you to handle or solve matrix equations, predict system behavior, and apply them to differential equations.

Systems of Equations

Differential equations often involve systems of equations, especially when modeling interconnected phenomena, like electrical circuits or population dynamics. Breaking these down into manageable components is crucial for solving them effectively.
Here’s how to understand systems of equations:

• Each equation in the system represents a relationship between the variables, often dependent on time or another parameter.
• The goal is to find solutions for all variables that satisfy every equation simultaneously.
• When expressing systems in matrix form, you systematically reduce complexity by using vectors and matrices, as demonstrated in our exercise.

Using the matrix form of a system of differential equations can provide insights into their stability and solutions. It allows for analysis and computation using advanced mathematical tools, offering a clearer understanding of the system's dynamics.