Problem 8 - Advanced Engineering Mathematics

Question

In Problems 7-10, write the given system without the use of matrices. $$ \mathbf{X}^{\prime}=\left(\begin{array}{rrr} 7 & 5 & -9 \\ 4 & 1 & 1 \\ 0 & -2 & 3 \end{array}\right) \mathbf{X}+\left(\begin{array}{l} 0 \\ 2 \\ 1 \end{array}\right) e^{5 t}-\left(\begin{array}{l} 8 \\ 0 \\ 3 \end{array}\right) e^{-2 t} $$

Step-by-Step Solution

Answer

The system is: 1) \( x_1' = 7x_1 + 5x_2 - 9x_3 - 8e^{-2t} \), 2) \( x_2' = 4x_1 + x_2 + x_3 + 2e^{5t} \), 3) \( x_3' = -2x_2 + 3x_3 + e^{5t} - 3e^{-2t} \).

Step 1: Understanding the System of Differential Equations

This problem involves a system of linear differential equations represented in matrix form: \( \mathbf{X}' = A\mathbf{X} + \mathbf{B}e^{5t} - \mathbf{C}e^{-2t} \).Here, \( \mathbf{X} = \begin{pmatrix} x_1(t) \ x_2(t) \ x_3(t) \end{pmatrix} \) is a vector of dependent variables.The goal is to express this system without matrices.

Step 2: Extracting the System from the Matrix Form

In the given \( A = \begin{pmatrix} 7 & 5 & -9 \ 4 & 1 & 1 \ 0 & -2 & 3 \end{pmatrix}, \; \mathbf{B} = \begin{pmatrix} 0 \ 2 \ 1 \end{pmatrix}, \; \text{and} \; \mathbf{C} = \begin{pmatrix} 8 \ 0 \ 3 \end{pmatrix} \),we perform matrix multiplication and vector addition to find individual differential equations for \( x_1', x_2', \) and \( x_3' \).

Step 3: Expanding the First Equation

Calculate the first equation: \[ x_1' = (7x_1 + 5x_2 - 9x_3) + 0 \cdot e^{5t} - 8 \cdot e^{-2t} \] This simplifies to: \[ x_1' = 7x_1 + 5x_2 - 9x_3 - 8e^{-2t} \].

Step 4: Expanding the Second Equation

For the second equation: \[ x_2' = (4x_1 + 1x_2 + 1x_3) + 2e^{5t} + 0 \cdot e^{-2t} \] This simplifies to: \[ x_2' = 4x_1 + x_2 + x_3 + 2e^{5t} \].

Step 5: Expanding the Third Equation

For the third equation: \[ x_3' = (0x_1 - 2x_2 + 3x_3) + 1e^{5t} - 3e^{-2t} \] This simplifies to: \[ x_3' = -2x_2 + 3x_3 + e^{5t} - 3e^{-2t} \].

Key Concepts

Matrix Form

In the world of differential equations, expressing a system in matrix form can drastically simplify the problem-solving process. The given problem is initially presented using matrices, where a matrix is essentially an organized collection of numbers arranged into rows and columns.

For systems of linear differential equations, the matrix form enables us to compactly outline several equations into a single, cohesive statement. This is particularly advantageous for dealing with multiple dependent variables at once. The system of equations is represented as:

• \( \mathbf{X}' = A\mathbf{X} + \mathbf{B}e^{5t} - \mathbf{C}e^{-2t} \)

Here, \( A \) is the coefficient matrix, \( \mathbf{X} \) is the vector of dependent variables, and \( \mathbf{B} \) and \( \mathbf{C} \) are vectors involved in the differential equations.

Breaking down this matrix form to relate it to individual differential equations helps us understand and solve the system more effectively. This structured approach makes handling complex relationships within different variables more manageable.

Differential Equations Expansion

When converting a system from matrix form to non-matrix equations, we perform a process known as expanding the differential equations. Each row in the coefficient matrix corresponds to a differential equation based on the dependent variables' derivatives.

The expansion involves multiplying and summing elements across each row of the coefficient matrix \( A \) with corresponding components of \( \mathbf{X} \), and then adding or subtracting influences from \( \mathbf{B}e^{5t} \) and \( \mathbf{C}e^{-2t} \). This is how we arrive at three separate differential equations:

• For the first row:
  \( x_1' = 7x_1 + 5x_2 - 9x_3 - 8e^{-2t} \)
• For the second row:
  \( x_2' = 4x_1 + x_2 + x_3 + 2e^{5t} \)
• For the third row:
  \( x_3' = -2x_2 + 3x_3 + e^{5t} - 3e^{-2t} \)

Each of these equations expands one component of the system, expressing the rate of change of each variable in terms of all variables involved. Understanding this expansion is crucial for handling systems of linear differential equations without matrices.

Dependent Variables

In systems of linear differential equations, dependent variables are the core elements that we solve for. These variables, denoted often as \( x_1(t), x_2(t), \) and \( x_3(t) \), represent quantities that change over time according to the differential equations provided in the problem.

Recognizing these as dependent variables signifies that their values depend on other variables, time (\( t \)), and the nature of the system itself. Each equation in our expanded set relates these variables with their derivatives. These relationships are critical in understanding how the system evolves over time.

Taking a closer look:

• \( x_1(t) \) depends on the calculations and influences from \( x_2(t), x_3(t), e^{-2t} \).
• Similarly, \( x_2(t) \) is shaped by \( x_1(t), x_3(t), e^{5t} \), indicating the interconnected nature of these variables.
• \( x_3(t) \) shows dependency through \( x_2(t) \), its unique combination with the functions of \( e^{5t} \), and \( e^{-2t} \).

By fully grasping the concept of dependent variables, one can better comprehend the underlying dynamics of linear differential equations, making the solution process more intuitive and structured.