21 = 0 with . and = 4 with v2 = Write the solution to the linear system r' = Ar in the following forms. A. In eigenvalue/eigenvector form: x(t) = C1 + C2 y(t) e e B. In fundamental matrix form: - x(t) C1 y(t). C2 C. As two equations: (write "c1" and "c2" for ci and c2 ) x(t) = y(t) =
21 = 0 with . and = 4 with v2 = Write the solution to the linear system r' = Ar in the following forms. A. In eigenvalue/eigenvector form: x(t) = C1 + C2 y(t) e e B. In fundamental matrix form: - x(t) C1 y(t). C2 C. As two equations: (write "c1" and "c2" for ci and c2 ) x(t) = y(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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Question
Suppose that the matrix AA has the following eigenvalues and eigenvectors:
![Transcription for Educational Website:
---
**Eigenvalue and Eigenvector Analysis**
Given the eigenvalues and eigenvectors:
- \( \lambda_1 = 0 \) with \( \mathbf{v}_1 = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \).
- \( \lambda_2 = 4 \) with \( \mathbf{v}_2 = \begin{bmatrix} 0 \\ 3 \end{bmatrix} \).
Write the solution to the linear system \( \mathbf{r}' = A\mathbf{r} \) in the following forms:
**A. In eigenvalue/eigenvector form:**
\[
\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = c_1 \begin{bmatrix} \Box \\ \Box \end{bmatrix} e^{\Box t} + c_2 \begin{bmatrix} \Box \\ \Box \end{bmatrix} e^{\Box t}
\]
**B. In fundamental matrix form:**
\[
\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = \begin{bmatrix} \Box & \Box \\ \Box & \Box \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}
\]
**C. As two equations: (write "c1" and "c2" for \( c_1 \) and \( c_2 \))**
\[
x(t) = \Box
\]
\[
y(t) = \Box
\]
---
Replace each \( \Box \) with the appropriate expression or value derived from solving the linear system with the given eigenvalues and eigenvectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2eee5fef-24c0-42d2-a99f-ec7ca4df2e97%2F8ca92cb0-31a3-46e6-be62-df2720a8131f%2Fej6hma_processed.png&w=3840&q=75)
Transcribed Image Text:Transcription for Educational Website:
---
**Eigenvalue and Eigenvector Analysis**
Given the eigenvalues and eigenvectors:
- \( \lambda_1 = 0 \) with \( \mathbf{v}_1 = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \).
- \( \lambda_2 = 4 \) with \( \mathbf{v}_2 = \begin{bmatrix} 0 \\ 3 \end{bmatrix} \).
Write the solution to the linear system \( \mathbf{r}' = A\mathbf{r} \) in the following forms:
**A. In eigenvalue/eigenvector form:**
\[
\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = c_1 \begin{bmatrix} \Box \\ \Box \end{bmatrix} e^{\Box t} + c_2 \begin{bmatrix} \Box \\ \Box \end{bmatrix} e^{\Box t}
\]
**B. In fundamental matrix form:**
\[
\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = \begin{bmatrix} \Box & \Box \\ \Box & \Box \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}
\]
**C. As two equations: (write "c1" and "c2" for \( c_1 \) and \( c_2 \))**
\[
x(t) = \Box
\]
\[
y(t) = \Box
\]
---
Replace each \( \Box \) with the appropriate expression or value derived from solving the linear system with the given eigenvalues and eigenvectors.
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