Problem 4 - Calculus for Biology and Medicine

Question

Write each system of differential equations in matrix form. \(\begin{aligned} & \frac{d x_{1}}{d t}=2 x_{2}-3 x_{1}-x_{3} \\ & \frac{d x_{2}}{d t}=x_{2}-2 x_{1} \\ & \frac{d x_{3}}{d t}=5 x_{1}+x_{3} \end{aligned}\)

Step-by-Step Solution

Answer

The system in matrix form is: \( \frac{d}{dt} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} -3 & 2 & -1 \\ -2 & 1 & 0 \\ 5 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}. \)

Step 1: Identify Components

Identify the variables and their derivatives in the given system of differential equations. We have variables \(x_1, x_2, x_3\) with their respective derivatives \(\frac{dx_1}{dt}, \frac{dx_2}{dt}, \frac{dx_3}{dt}\).

Step 2: Express in Matrix Form

Write the system of equations in matrix form. This involves expressing the derivatives as a vector and the coefficients of each \(x\) variable as a matrix. The equation becomes: \[ \frac{d}{dt} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} -3 & 2 & -1 \ -2 & 1 & 0 \ 5 & 0 & 1 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix}. \]

Step 3: Verify the Matrix Equation

Ensure that the matrix multiplication on the right-hand side recreates the original system of differential equations. For example: - The first equation becomes \( -3x_1 + 2x_2 - x_3 \), matching \( \frac{dx_1}{dt} = 2x_2 - 3x_1 - x_3 \). - The second equation becomes \( -2x_1 + x_2 \), matching \( \frac{dx_2}{dt} = x_2 - 2x_1 \). - The third equation becomes \( 5x_1 + x_3 \), matching \( \frac{dx_3}{dt} = 5x_1 + x_3 \). This shows the matrix form is correct.

Key Concepts

Matrix Form

Matrix form is a powerful way to represent systems of equations, especially differential equations. It helps simplify problems, making them easier to manage and solve. When you convert a system of equations into matrix form, you're basically grouping coefficients into a matrix and variables into a vector.

For the given system of differential equations, the equation is written as a matrix where the columns correspond to coefficients of each variable. This allows for concise notation and easier computation, especially for large systems:

• Each equation's left-hand side, an element, is a derivative of a variable.
• On the right-hand side, you'll have the variables with their respective coefficients in matrix format.
• This results in a matrix multiplication, where each row of the matrix represents one differential equation.

To convert a system into matrix form, first identify the coefficient of each variable across the equations. Arrange these coefficients systematically in a square grid-like structure, which is your matrix. Let the vector represent the variables whose derivatives you are considering. This structured approach makes analyzing the behavior and solutions of larger systems much easier.

System of Equations

A system of equations involves multiple equations that share variables. These systems can be linear or differential, consisting of functions or their derivatives.

With linear equations, you'll often see them written individually. However, differential equations often benefit from being in a matrix form, which succinctly encapsulates the relationship between the rates of change and the variables.

When tackling a system of differential equations, the key is to first express each variable’s rate of change with respect to time (or other variables). In our provided exercise:

• The system has three equations where each derivative is expressed as a linear combination of all the variables.
• These can be treated simultaneously once put in matrix form, leveraging linear algebra techniques for solving.
• Working with systems this way means that you can often predict and control the behavior of dynamic systems in mechanical, electrical, or even biological contexts.

Approaching such problems with matrices allows for clear visualization and analysis of how each variable influences the others and changes overtime.

Derivatives

Derivatives represent rates of change and are a cornerstone of calculus and differential equations. They gauge how a function changes as its input changes.

In our system of differential equations, derivatives are used to express how each variable changes over time. This concept helps in modeling real-world phenomena where change is prominent, such as population growth, heat transfer, or movement of objects.

For instance:

• The derivative \( \frac{dx_1}{dt} \) shows how the variable \( x_1 \) evolves with time; here influenced by the values of \( x_2 \), \( x_3 \), and \( x_1 \) itself.
• Each derivative is a linear function of the system’s variables, making the whole system a set of linear differential equations.
• Understanding how derivatives unfold within these equations lets us solve for unknown functions that describe the system's behavior over time.

By working with derivatives in systems of differential equations, you can predict future behavior, translate physical laws into mathematical terms, and drive innovations in engineering and technology fields.