In Exercises 17–19, each of the given linear systems has zero as an eigenvalue. For each system, (a) find the eigenvalues; (b) find the eigenvectors; (c) sketch the phase portrait; (d) sketch the x(t)- and y(t)-graphs of the solution with initial condition Yo = (1, 0); %3D (e) find the general solution; and (f) find the particular solution for the initial condition Yo : your sketch from part (d). (1, 0) and compare it with n. - (:) -(:) -() dY 17. dt 4 2 Y 2 1 2 dY 18. dt 2 4 Y 3 6 dY 19. dt 0 -1
In Exercises 17–19, each of the given linear systems has zero as an eigenvalue. For each system, (a) find the eigenvalues; (b) find the eigenvectors; (c) sketch the phase portrait; (d) sketch the x(t)- and y(t)-graphs of the solution with initial condition Yo = (1, 0); %3D (e) find the general solution; and (f) find the particular solution for the initial condition Yo : your sketch from part (d). (1, 0) and compare it with n. - (:) -(:) -() dY 17. dt 4 2 Y 2 1 2 dY 18. dt 2 4 Y 3 6 dY 19. dt 0 -1
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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Can you please help me solve questions 17 and 19?
![**Exercises 17–19: Linear Systems with Zero Eigenvalue**
For each linear system provided below, perform the following tasks:
(a) Determine the eigenvalues.
(b) Find the eigenvectors corresponding to the eigenvalues.
(c) Sketch the phase portrait of the system.
(d) Sketch the graphs of \(x(t)\) and \(y(t)\) for the solution with the initial condition \( \mathbf{Y}_0 = (1, 0) \).
(e) Determine the general solution of the system.
(f) Find the particular solution for the initial condition \( \mathbf{Y}_0 = (1, 0) \) and compare it with your sketches from part (d).
**Exercise 17:**
\[ \frac{d\mathbf{Y}}{dt} = \begin{pmatrix} 0 & 2 \\ 0 & -1 \end{pmatrix} \mathbf{Y} \]
**Exercise 18:**
\[ \frac{d\mathbf{Y}}{dt} = \begin{pmatrix} 2 & 4 \\ 3 & 6 \end{pmatrix} \mathbf{Y} \]
**Exercise 19:**
\[ \frac{d\mathbf{Y}}{dt} = \begin{pmatrix} 4 & 2 \\ 2 & 1 \end{pmatrix} \mathbf{Y} \]
**Note:** Each system has zero as one of its eigenvalues.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed967076-81f7-4f42-89f7-9fef1520b813%2F49a354a0-a4f2-4593-8d32-4be516bbd680%2Fns81bsb_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercises 17–19: Linear Systems with Zero Eigenvalue**
For each linear system provided below, perform the following tasks:
(a) Determine the eigenvalues.
(b) Find the eigenvectors corresponding to the eigenvalues.
(c) Sketch the phase portrait of the system.
(d) Sketch the graphs of \(x(t)\) and \(y(t)\) for the solution with the initial condition \( \mathbf{Y}_0 = (1, 0) \).
(e) Determine the general solution of the system.
(f) Find the particular solution for the initial condition \( \mathbf{Y}_0 = (1, 0) \) and compare it with your sketches from part (d).
**Exercise 17:**
\[ \frac{d\mathbf{Y}}{dt} = \begin{pmatrix} 0 & 2 \\ 0 & -1 \end{pmatrix} \mathbf{Y} \]
**Exercise 18:**
\[ \frac{d\mathbf{Y}}{dt} = \begin{pmatrix} 2 & 4 \\ 3 & 6 \end{pmatrix} \mathbf{Y} \]
**Exercise 19:**
\[ \frac{d\mathbf{Y}}{dt} = \begin{pmatrix} 4 & 2 \\ 2 & 1 \end{pmatrix} \mathbf{Y} \]
**Note:** Each system has zero as one of its eigenvalues.
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