Problem 8 - Advanced Engineering Mathematics

Question

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

Step 1: Express the System of Differential Equations

First, identify the components of the matrix equation \( \mathbf{X}^{\prime}=A\mathbf{X}+\mathbf{B}\). Here, \( A \) is the coefficient matrix, and \( \mathbf{B} \) is the function of \( t \). If \( \mathbf{X} = \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} \), then \( \mathbf{X}^{\prime} = \begin{pmatrix} x_1' \ x_2' \ x_3' \end{pmatrix} \).

Step 2: Multiply Matrix A by X

Calculate \( A \mathbf{X} \) resulting in:\[ \begin{pmatrix} 7 & 5 & -9 \ 4 & 1 & 1 \ 0 & -2 & 3 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \ x_3 \end{pmatrix} = \begin{pmatrix} 7x_1 + 5x_2 - 9x_3 \ 4x_1 + x_2 + x_3 \ -2x_2 + 3x_3 \end{pmatrix} \].

Step 3: Express Function B

We have \( \mathbf{B}(t) = \begin{pmatrix} 0 \ 2 \ 1 \end{pmatrix} e^{5t} - \begin{pmatrix} 8 \ 0 \ 3 \end{pmatrix} e^{-2t} \). Compute this to get:\[ \mathbf{B}(t) = \begin{pmatrix} 0 \ 2e^{5t} \ e^{5t} \end{pmatrix} - \begin{pmatrix} 8e^{-2t} \ 0 \ 3e^{-2t} \end{pmatrix} = \begin{pmatrix} -8e^{-2t} \ 2e^{5t} \ e^{5t} - 3e^{-2t} \end{pmatrix} \].

Step 4: Combine the Steps to Form the System

Substitute the results from Step 2 and Step 3 into the system:\[\begin{align*}x_1' &= 7x_1 + 5x_2 - 9x_3 - 8e^{-2t}, \x_2' &= 4x_1 + x_2 + x_3 + 2e^{5t}, \x_3' &= -2x_2 + 3x_3 + e^{5t} - 3e^{-2t}.\end{align*}\]

Step 5: Finalize the System of Equations

The complete system of differential equations without using matrices is:\[\begin{aligned}x_1' &= 7x_1 + 5x_2 - 9x_3 - 8e^{-2t}, \x_2' &= 4x_1 + x_2 + x_3 + 2e^{5t}, \x_3' &= -2x_2 + 3x_3 + e^{5t} - 3e^{-2t}.\end{aligned}\]

Key Concepts

System of Differential Equations

A system of differential equations involves multiple equations containing derivatives of more than one dependent variable with respect to one or more independent variables. These systems are crucial in understanding how several phenomena evolve simultaneously. For your exercise, the solution aims to write a system of equations in its non-matrix form. This is important because it provides a more direct representation of the relationships between variables without the abstraction of matrices.

• The system is represented by three equations involving derivatives of three variables.
• Each equation describes the rate of change of one variable in terms of others.
• This format can be useful for scenarios with multiple interacting factors, such as in physics and engineering.

Grasping this concept offers insight into the dynamics of complex systems and aids in learning how changes in certain inputs affect the system as a whole.

Matrix Equations

Matrix equations are a powerful tool for handling systems of linear equations. They allow representation of multiple linear equations in a compact and concise form, making them easier to manipulate mathematically. In your given exercise, each equation is transformed into a matrix format enabling a clearer formulation and solution approach.

• The coefficients of the variables form a matrix, facilitating computation through operations like multiplication and inversion.
• By expressing the system in matrix terms, solving the system can become more efficient.
• Using matrices provides a path to apply linear algebra techniques such as eigenvalues and eigenvectors for further analysis.

Understanding matrix equations is beneficial because they provide a structured way to deal with complex, multi-variable problems simply and effectively.

Transformation of Equations

Transformation of equations refers to altering equations from one form to another to make them easier to solve or analyze. In the exercise, you begin by identifying components, performing matrix operations, and then combining results to transform the system. This step-by-step transformation from matrix to non-matrix form aids clarity and understanding.

• This process often involves breaking down complex forms into simpler parts.
• Transformation can provide greater insight into the behavior of each component of the system.
• It simplifies solving by focusing on individual aspects rather than the entire system at once.

Through transformation, problems become more approachable, and solutions more evident, reducing the potential for missteps and confusion.