The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system. x₁ = 6x₁ +5x₂ + 5x3, x2 = -7x₁ - 6x₂ - 5x3, x'3 = 7x₁ + 7x₂ + 6x3 What is the general solution in matrix form? x(t) =
The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system. x₁ = 6x₁ +5x₂ + 5x3, x2 = -7x₁ - 6x₂ - 5x3, x'3 = 7x₁ + 7x₂ + 6x3 What is the general solution in matrix form? x(t) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Solving Systems of Linear Differential Equations Using Eigenvalues**
The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system.
\[
x'_1 = 6x_1 + 5x_2 + 5x_3, \quad x'_2 = -7x_1 - 6x_2 - 5x_3, \quad x'_3 = 7x_1 + 7x_2 + 6x_3
\]
---
**Problem Statement**
What is the general solution in matrix form?
\[
\mathbf{x}(t) = \quad \boxed{\phantom{answer}}
\]
To find the general solution:
1. **Form the Coefficient Matrix**: Construct the matrix A of the system of differential equations.
2. **Find Eigenvalues**: Determine the eigenvalues of matrix A.
3. **Find Eigenvectors**: Calculate the eigenvectors corresponding to each eigenvalue.
4. **Construct General Solution**: Form the general solution using the eigenvalues and eigenvectors.
By completing these steps, you will derive the general matrix form solution for the system provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06bbd155-c9d1-4bdc-8881-0cd893ac016a%2F2db9ac11-3bff-4e15-8424-0b54e5cf2f10%2Ff69k0h3_processed.png&w=3840&q=75)
Transcribed Image Text:**Solving Systems of Linear Differential Equations Using Eigenvalues**
The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system.
\[
x'_1 = 6x_1 + 5x_2 + 5x_3, \quad x'_2 = -7x_1 - 6x_2 - 5x_3, \quad x'_3 = 7x_1 + 7x_2 + 6x_3
\]
---
**Problem Statement**
What is the general solution in matrix form?
\[
\mathbf{x}(t) = \quad \boxed{\phantom{answer}}
\]
To find the general solution:
1. **Form the Coefficient Matrix**: Construct the matrix A of the system of differential equations.
2. **Find Eigenvalues**: Determine the eigenvalues of matrix A.
3. **Find Eigenvectors**: Calculate the eigenvectors corresponding to each eigenvalue.
4. **Construct General Solution**: Form the general solution using the eigenvalues and eigenvectors.
By completing these steps, you will derive the general matrix form solution for the system provided.
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