Problem 5 - Advanced Engineering Mathematics

Question

The system is $$\begin{aligned}&x_{1}^{\prime}=2 \cdot 3+\frac{1}{50} x_{2}-\frac{1}{50} x_{1} \cdot 4=-\frac{2}{25} x_{1}+\frac{1}{50} x_{2}+6\\\&x_{2}^{\prime}=\frac{1}{50} x_{1} \cdot 4-\frac{1}{50} x_{2}-\frac{1}{50} x_{2} \cdot 3=\frac{2}{25} x_{1}-\frac{2}{25} x_{2}\end{aligned}$$

Step-by-Step Solution

Answer

The system simplifies to a matrix form: \[ \begin{bmatrix} x_1' \\ x_2' \end{bmatrix} = \begin{bmatrix} -\frac{2}{25} & \frac{1}{50} \\ \frac{2}{25} & -\frac{2}{25} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 6 \\ 0 \end{bmatrix} \].

Step 1: Simplify Equations

First, let's simplify the given equations. The first equation, \(x_1' = 2 \cdot 3 + \frac{1}{50} x_2 - \frac{1}{50} x_1 \cdot 4\), simplifies to \(-\frac{2}{25} x_1 + \frac{1}{50} x_2 + 6\). The second equation, \(x_2' = \frac{1}{50} x_1 \cdot 4 - \frac{1}{50} x_2 - \frac{1}{50} x_2 \cdot 3\), simplifies to \(\frac{2}{25} x_1 - \frac{2}{25} x_2\). After combining and simplifying terms, these equations are as follows.

Step 2: Identify Coefficients and Constants

For the first equation: \(x_1' = -\frac{2}{25} x_1 + \frac{1}{50}x_2 + 6\). Here, the coefficient of \(x_1\) is \(-\frac{2}{25}\), the coefficient of \(x_2\) is \(\frac{1}{50}\), and the constant term is \(6\). For the second equation: \(x_2' = \frac{2}{25} x_1 - \frac{2}{25} x_2\). Here, the coefficient of \(x_1\) is \(\frac{2}{25}\), and the coefficient of \(x_2\) is \(-\frac{2}{25}\). There is no constant term.

Step 3: Write Matrix Form of the System

Represent the system with a matrix. The system can be represented as \(\begin{bmatrix} x_1' \ x_2' \end{bmatrix} = \begin{bmatrix} -\frac{2}{25} & \frac{1}{50} \ \frac{2}{25} & -\frac{2}{25} \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} + \begin{bmatrix} 6 \ 0 \end{bmatrix}\). This matrix equation separates the linear terms from the constant terms.

Step 4: Understand the System Dynamics

The matrix form tells us that the change in \(x_1\) depends negatively on itself and slightly positively on \(x_2\), while having a constant inflow. The change in \(x_2\) depends positively on \(x_1\) and negatively on itself. This sets up the system dynamics for analysis and potential finding of equilibrium or behavior over time.

Key Concepts

Matrix Formulation

Linear differential equations can often be represented in matrix form, which provides clarity and efficiency when solving. In this exercise, we have a system of equations, each describing the rate of change of a variable. The matrix formulation helps us aggregate these equations into a single operator. This matrix, known as the coefficient matrix, contains the coefficients of each variable as elements.

By transposing our system of equations into a matrix, we can utilize linear algebra tools to solve or analyze them. The matrix representation is given by:

• A coefficient matrix that aligns each term of the equations.
• A vector of variables, indicating how each variable connects to the system.
• A constant vector that separates constant terms from variable-dependent parts.

This structured approach simplifies understanding complex systems by focusing on a matrix instead of multiple detailed equations. This matrix consists of rows and columns representing how each variable impacts others directly.

System of Equations

At the core of this exercise lies a system of linear differential equations. These systems are sets of equations that describe how variables change over time.

Each equation provides insight into the dynamics affecting a particular variable. For example, the equations given initially express the rates of change of two variables, \(x_1\) and \(x_2\). Systems of such nature are common in modeling real-world situations where quantities evolve at rates dependent on each other and on external inputs.

In these equations:

• \(x_1'\) and \(x_2'\) are the derivatives, representing the rate of change of \(x_1\) and \(x_2\), respectively.
• The system captures not only how each variable changes, but also how they influence each other.

Understanding these interactions is crucial for solving the system or predicting future states of the variables.

Simplification of Equations

Simplifying the original system of equations is a necessary step before attempting to further analyze or solve them. This involves rewriting the equations to reduce complexity and remove redundancies.

In our original example, the terms are recombined to improve clarity:

• The first equation simplifies from multiple terms into \(-\frac{2}{25} x_1 + \frac{1}{50} x_2 + 6\).
• The second equation reorganizes to \(\frac{2}{25} x_1 - \frac{2}{25} x_2\).

These simplifications do not alter the system's behavior but make it more manageable by focusing on how coefficients and terms interact directly. It also aids in performing operations like finding solutions or dynamical system analyses.

Coefficient Identification

Identifying coefficients is one of the initial steps in tackling any system of linear equations. Coefficients are the numerical or algebraic factors that multiply variables in each equation.

By parsing the simplified equations:

• The first equation has coefficients \(-\frac{2}{25}\) for \(x_1\) and \(\frac{1}{50}\) for \(x_2\), with a constant term of \(6\).
• The second equation reveals coefficients \(\frac{2}{25}\) for \(x_1\) and \(-\frac{2}{25}\) for \(x_2\), without a constant term.

Recognizing these coefficients is critical for many tasks:

• Facilitating the writing of the matrix representation.
• Determining how each variable dynamically interacts within the system.
• Enabling direct comparison among different systems.

In the context of a dynamic system, identifying these parameters helps predict future behavior or tune system responses to desired levels.