The eigenvalues of the coefficient matrix A are given. Find a general solution of the indicated system x' = Ax. 77 16 -32 -72 - 11 32 x; λ=-3, 5, 5 144 32 - 59 x(t) =
The eigenvalues of the coefficient matrix A are given. Find a general solution of the indicated system x' = Ax. 77 16 -32 -72 - 11 32 x; λ=-3, 5, 5 144 32 - 59 x(t) =
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Statement on Eigenvalues and Differential Equations**
The eigenvalues of the coefficient matrix \( A \) are given. Find a general solution of the indicated system \( \mathbf{x}' = A \mathbf{x} \).
Given the matrix \( A \) and the vector \( \mathbf{x} \):
\[
\mathbf{x}' = \begin{bmatrix}
77 & 16 & -32 \\
-72 & -11 & 32 \\
144 & 32 & -59
\end{bmatrix} \mathbf{x}
\]
where the eigenvalues \( \lambda \) are \( -3, 5, 5 \).
**Solution Approach:**
To find the general solution of the system \( \mathbf{x}' = A \mathbf{x} \), where the matrix \( A \) has the eigenvalues \( -3, 5, 5 \), follow these steps:
1. **Eigenvector Calculation:**
- Determine the eigenvectors associated with each eigenvalue.
2. **General Solution Form:**
- The general solution to the system of differential equations is a linear combination of the solutions associated with each eigenvalue, where each solution is of the form \( \mathbf{x}(t) = \mathbf{v} e^{\lambda t} \).
- Since the eigenvalue \( 5 \) is repeated, you may also need to find a generalized eigenvector if \( \mathbf{v}_2 \) and \( \mathbf{v}_3 \) are not linearly independent.
Without further calculations, the general solution can be represented as:
\[
\mathbf{x}(t) = c_1 \mathbf{v}_1 e^{-3t} + c_2 \mathbf{v}_2 e^{5t} + c_3 \mathbf{v}_3 e^{5t}
\]
where \( \mathbf{v}_1 \), \( \mathbf{v}_2 \), and \( \mathbf{v}_3 \) are the eigenvectors associated with \( \lambda = -3 \) and \( \lambda = 5 \) respectively, and \( c_1 \), \( c_2 \), \( c_3 \) are constants determined by the initial conditions.
**Explanation of the Anticipated Diagrams or Graphs:**
- **Eigenvector Matrix:** A matrix diagram illustrating how to compute the eigenvectors from the eigen](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F896deee6-4ebc-4afd-8502-502eb7aa6712%2F5b85df72-72ae-428d-ac67-09b661e36ab3%2Fvcjgn1o_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement on Eigenvalues and Differential Equations**
The eigenvalues of the coefficient matrix \( A \) are given. Find a general solution of the indicated system \( \mathbf{x}' = A \mathbf{x} \).
Given the matrix \( A \) and the vector \( \mathbf{x} \):
\[
\mathbf{x}' = \begin{bmatrix}
77 & 16 & -32 \\
-72 & -11 & 32 \\
144 & 32 & -59
\end{bmatrix} \mathbf{x}
\]
where the eigenvalues \( \lambda \) are \( -3, 5, 5 \).
**Solution Approach:**
To find the general solution of the system \( \mathbf{x}' = A \mathbf{x} \), where the matrix \( A \) has the eigenvalues \( -3, 5, 5 \), follow these steps:
1. **Eigenvector Calculation:**
- Determine the eigenvectors associated with each eigenvalue.
2. **General Solution Form:**
- The general solution to the system of differential equations is a linear combination of the solutions associated with each eigenvalue, where each solution is of the form \( \mathbf{x}(t) = \mathbf{v} e^{\lambda t} \).
- Since the eigenvalue \( 5 \) is repeated, you may also need to find a generalized eigenvector if \( \mathbf{v}_2 \) and \( \mathbf{v}_3 \) are not linearly independent.
Without further calculations, the general solution can be represented as:
\[
\mathbf{x}(t) = c_1 \mathbf{v}_1 e^{-3t} + c_2 \mathbf{v}_2 e^{5t} + c_3 \mathbf{v}_3 e^{5t}
\]
where \( \mathbf{v}_1 \), \( \mathbf{v}_2 \), and \( \mathbf{v}_3 \) are the eigenvectors associated with \( \lambda = -3 \) and \( \lambda = 5 \) respectively, and \( c_1 \), \( c_2 \), \( c_3 \) are constants determined by the initial conditions.
**Explanation of the Anticipated Diagrams or Graphs:**
- **Eigenvector Matrix:** A matrix diagram illustrating how to compute the eigenvectors from the eigen
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