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.

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Description: 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

Hello. Since your question has multiple parts, we will solve the first part for you. If you want remaining sub-parts to be solved, then please resubmit the whole question and specify those part you want us to solve. 17Consider a system of linear differential equation:dYdt=020-1YTo find:aThe eigen valuesbThe eigen vectorscSketch the phase portrait:Consider a system of linear differential equation:dYdt=020-1YLetA=020-1To calculate:A-λI=00-λ20-1-λ=0-λ-1-λ=0⇒λ=0,-1All eigen values are distinct and real,unequal and of the opposite sign.Phase line is unstable and has saddle points.bTo find eigen vector corrosponding the eigen values λ=0,-1.Eigen vector corrosponding to eigen value λ=0 is020-1y1y2=02y2=0 and -y2=0Then,eigen vector v1 corrosponding to eigen value λ=0 is v1=10.Eigen vector corrosponding to eigen value λ=-1 is-1200y1y2=0-y1+2y2=0Put y2=c,then y1=2c, thenEigen vectorv2 corrosponding to eigen value λ=-1 is v2=21.(c) Sketching of phase portrait will be

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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.

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