Let y' = Ay be a system of differential equations where A = 0 The matrix has spectrum X(A) = {-4} and c = -4 [+] The vector d = [4] satisfies the equation (A - rI)d = c. What is the general solution to the system of differential equations? is an eigenvector of A corresponding to r = -4. 4 Yı Ex: 6 t Ex: 6 = ke=²² [²] + k₂ (to 13) k₁e te Y2 t te
Let y' = Ay be a system of differential equations where A = 0 The matrix has spectrum X(A) = {-4} and c = -4 [+] The vector d = [4] satisfies the equation (A - rI)d = c. What is the general solution to the system of differential equations? is an eigenvector of A corresponding to r = -4. 4 Yı Ex: 6 t Ex: 6 = ke=²² [²] + k₂ (to 13) k₁e te Y2 t te
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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linear algbra problem
Systems of differential equations with repeated real eigenvalues.
![Let \(\mathbf{y'} = A\mathbf{y}\) be a system of differential equations where \( A = \begin{bmatrix} -4 & 4 \\ 0 & -4 \end{bmatrix} \).
The matrix has spectrum \(\lambda(A) = \{-4\}\) and \(\mathbf{c} = \begin{bmatrix} -4 \\ 0 \end{bmatrix}\) is an eigenvector of \(A\) corresponding to \(r = -4\).
The vector \(\mathbf{d} = \begin{bmatrix} 4 \\ -1 \end{bmatrix}\) satisfies the equation \((A - rI)\mathbf{d} = \mathbf{c}\).
What is the general solution to the system of differential equations?
\[
\begin{bmatrix} y_1 \\ y_2 \end{bmatrix} = k_1 e^{-4t} \begin{bmatrix} -4 \\ 0 \end{bmatrix} + k_2 \left( t e^{-4t} \begin{bmatrix} -4 \\ 0 \end{bmatrix} + e^{-4t} \begin{bmatrix} 4 \\ -1 \end{bmatrix} \right)
\]
The image includes a matrix, vector, and linear combination representing the solution to the differential equations. The solution components use exponential functions and constants, illustrating the general structure of solving linear differential systems with repeated eigenvalues.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab04051d-220f-4e22-ad42-dacf93a25243%2F49440f4f-fda5-41e5-affc-4539c9f5a46d%2Fkw8fl04_processed.png&w=3840&q=75)
Transcribed Image Text:Let \(\mathbf{y'} = A\mathbf{y}\) be a system of differential equations where \( A = \begin{bmatrix} -4 & 4 \\ 0 & -4 \end{bmatrix} \).
The matrix has spectrum \(\lambda(A) = \{-4\}\) and \(\mathbf{c} = \begin{bmatrix} -4 \\ 0 \end{bmatrix}\) is an eigenvector of \(A\) corresponding to \(r = -4\).
The vector \(\mathbf{d} = \begin{bmatrix} 4 \\ -1 \end{bmatrix}\) satisfies the equation \((A - rI)\mathbf{d} = \mathbf{c}\).
What is the general solution to the system of differential equations?
\[
\begin{bmatrix} y_1 \\ y_2 \end{bmatrix} = k_1 e^{-4t} \begin{bmatrix} -4 \\ 0 \end{bmatrix} + k_2 \left( t e^{-4t} \begin{bmatrix} -4 \\ 0 \end{bmatrix} + e^{-4t} \begin{bmatrix} 4 \\ -1 \end{bmatrix} \right)
\]
The image includes a matrix, vector, and linear combination representing the solution to the differential equations. The solution components use exponential functions and constants, illustrating the general structure of solving linear differential systems with repeated eigenvalues.
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