Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Title: Solving a System of Differential Equations**
**Problem Statement:**
Solve the differential equation given by:
\[ \frac{dx}{dt} = \begin{bmatrix} 7 & 2 \\ -2 & 7 \end{bmatrix} x \]
with the initial condition:
\[ x(0) = \begin{bmatrix} 3 \\ 7 \end{bmatrix} \]
**Explanation:**
This problem involves solving a system of first-order linear differential equations represented in matrix form.
1. **Understanding the Matrix Differential Equation:**
The given equation can be expanded as:
\[
\frac{d}{dt} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 7 & 2 \\ -2 & 7 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}
\]
which represents a system of two coupled first-order differential equations.
2. **Initial Conditions:**
The initial condition is provided as:
\[
x(0) = \begin{bmatrix} 3 \\ 7 \end{bmatrix}
\]
This means that at time \( t = 0 \), the values of \( x_1 \) and \( x_2 \) are 3 and 7, respectively.
To solve this system, we typically use methods involving eigenvalues and eigenvectors of the coefficient matrix, or apply matrix exponentiation techniques.
**Graphical/Diagram Explanation:**
No graphs or diagrams accompany this problem. The problem purely involves algebraic manipulation and solution of a matrix differential equation.
**Educational Objective:**
By solving this problem, students will learn how to:
- Set up and interpret systems of differential equations in matrix form.
- Apply initial conditions to solve for specific solutions.
- Understand the methodology of solving coupled first-order linear differential equations using matrix algebra tools such as eigenvalues and eigenvectors.
The solution of such a problem also underpins crucial concepts in disciplines like control systems, dynamics, and other areas of applied mathematics and engineering.](https://content.bartleby.com/qna-images/question/f12b43ca-6b55-4575-a2c5-3f890463f057/9e7dc5d3-57cb-4a9d-8eb9-29aad2cba581/nreuapd_processed.jpeg)
Transcribed Image Text:**Title: Solving a System of Differential Equations**
**Problem Statement:**
Solve the differential equation given by:
\[ \frac{dx}{dt} = \begin{bmatrix} 7 & 2 \\ -2 & 7 \end{bmatrix} x \]
with the initial condition:
\[ x(0) = \begin{bmatrix} 3 \\ 7 \end{bmatrix} \]
**Explanation:**
This problem involves solving a system of first-order linear differential equations represented in matrix form.
1. **Understanding the Matrix Differential Equation:**
The given equation can be expanded as:
\[
\frac{d}{dt} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 7 & 2 \\ -2 & 7 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}
\]
which represents a system of two coupled first-order differential equations.
2. **Initial Conditions:**
The initial condition is provided as:
\[
x(0) = \begin{bmatrix} 3 \\ 7 \end{bmatrix}
\]
This means that at time \( t = 0 \), the values of \( x_1 \) and \( x_2 \) are 3 and 7, respectively.
To solve this system, we typically use methods involving eigenvalues and eigenvectors of the coefficient matrix, or apply matrix exponentiation techniques.
**Graphical/Diagram Explanation:**
No graphs or diagrams accompany this problem. The problem purely involves algebraic manipulation and solution of a matrix differential equation.
**Educational Objective:**
By solving this problem, students will learn how to:
- Set up and interpret systems of differential equations in matrix form.
- Apply initial conditions to solve for specific solutions.
- Understand the methodology of solving coupled first-order linear differential equations using matrix algebra tools such as eigenvalues and eigenvectors.
The solution of such a problem also underpins crucial concepts in disciplines like control systems, dynamics, and other areas of applied mathematics and engineering.
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