e) None of the above. 0 1 0 0 0 1 x is: he general solution of x' = -4 4 1 (1\ ( 1\
e) None of the above. 0 1 0 0 0 1 x is: he general solution of x' = -4 4 1 (1\ ( 1\
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Linear Algebra and Differential Equations: Solutions Analysis
#### Question 5: Fundamental Set of Solutions
Given the differential equation system:
\[
\mathbf{x}' = \begin{pmatrix} 0 & 1 \\ 10/t^2 & 4/t \end{pmatrix} \mathbf{x}
\]
Identify the fundamental set of solutions:
**Options:**
- (a) \(\mathbf{x}_1 = \begin{pmatrix} t^{-5} \\ t^2 \end{pmatrix}, \quad \mathbf{x}_2 = \begin{pmatrix} -5t^{-6} \\ 2t \end{pmatrix}\)
- (b) \(\mathbf{x}_1 = \begin{pmatrix} t^5 \\ 5t^4 \end{pmatrix}, \quad \mathbf{x}_2 = \begin{pmatrix} t^{-2} \\ -2t^{-3} \end{pmatrix}\)
- (c) \(\mathbf{x}_1 = \begin{pmatrix} t^5 \\ 5t^4 \end{pmatrix}, \quad \mathbf{x}_2 = \begin{pmatrix} t^2 \\ 2t \end{pmatrix}\)
- (d) \(\mathbf{x}_1 = \begin{pmatrix} t^{-5} \\ -5t^{-6} \end{pmatrix}, \quad \mathbf{x}_2 = \begin{pmatrix} t^2 \\ 2t \end{pmatrix}\)
- (e) None of the above
#### Question 6: General Solution
For the system:
\[
\mathbf{x}' = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -4 & 4 & 1 \end{pmatrix} \mathbf{x}
\]
Determine the general solution:
**Options:**
- (a) \(\mathbf{x} = C_1 e^{2t} \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} + C_2 e^{-2t} \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix} + C_3 e^t \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69d33b94-47ab-4124-8ded-71fa51390a3f%2F428bc448-f885-4fc9-ba9b-8206335f355f%2Ffctjawp_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Algebra and Differential Equations: Solutions Analysis
#### Question 5: Fundamental Set of Solutions
Given the differential equation system:
\[
\mathbf{x}' = \begin{pmatrix} 0 & 1 \\ 10/t^2 & 4/t \end{pmatrix} \mathbf{x}
\]
Identify the fundamental set of solutions:
**Options:**
- (a) \(\mathbf{x}_1 = \begin{pmatrix} t^{-5} \\ t^2 \end{pmatrix}, \quad \mathbf{x}_2 = \begin{pmatrix} -5t^{-6} \\ 2t \end{pmatrix}\)
- (b) \(\mathbf{x}_1 = \begin{pmatrix} t^5 \\ 5t^4 \end{pmatrix}, \quad \mathbf{x}_2 = \begin{pmatrix} t^{-2} \\ -2t^{-3} \end{pmatrix}\)
- (c) \(\mathbf{x}_1 = \begin{pmatrix} t^5 \\ 5t^4 \end{pmatrix}, \quad \mathbf{x}_2 = \begin{pmatrix} t^2 \\ 2t \end{pmatrix}\)
- (d) \(\mathbf{x}_1 = \begin{pmatrix} t^{-5} \\ -5t^{-6} \end{pmatrix}, \quad \mathbf{x}_2 = \begin{pmatrix} t^2 \\ 2t \end{pmatrix}\)
- (e) None of the above
#### Question 6: General Solution
For the system:
\[
\mathbf{x}' = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -4 & 4 & 1 \end{pmatrix} \mathbf{x}
\]
Determine the general solution:
**Options:**
- (a) \(\mathbf{x} = C_1 e^{2t} \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} + C_2 e^{-2t} \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix} + C_3 e^t \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix
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