Consider the coupled linear systemi = 3y – 4.xý = 5y – 6x1. Write this system in the formdæAæ.%3Ddt2. Find the eigenvalues and eigenvectors of A. What is the stability type of the origin?3. Find the general solution and draw a phase portrait (remember to label everything!)

Given: x˙=3y-4xy˙=5y-6x To find: 1) dxdt=Ax form of a given system. 2) Eigenvalues and Eigenvectors…

Answered: Consider the coupled linear system i =…

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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Multiple Choice) Consider the phase portrait below for the system x Ax, > 0 .4 -2 We would classify this system as a: We can deduce that the eigenvalues are: (a) stable node (a) of mixed sign (b) unstable node (b) both negative (c) both positive (c) saddle (d) spiral (d) Not enough information is given Write the solution as x(t) ceifv+czeatv2, where is the eigenvalue of largest magni- tude. Which of the following is va? (Briefly explain your answer below) (a) (1,-1) (b) (3, 1) (c) (1,3) (d) Not enough information is given 2.
the given figure is a phase portrait for a system x' = Ax, (a) The eigenvalues of A could be (circle one) i. A1 = 1, A2 = 2 ii. A1 = -1, A2 = -2 iii. A1 = -1, A2 = 2 iv. A1,2 = 1+i v. A1,2 = -1±i (b) If possible, give one eigenvector of A (if it is not possible, write "N/A"): %3D Answer:
6. The nonlinear system y₁ = -y₁ - y2, y₂ = y₁ + y₂, has steady states at (0,0), (1, -1), and (-1,1). Classify each steady state as a source (all eigenvalues have positive real part), sink (all eigenvalues have negative real part), saddle (one eigenvalue with positive real part and one with negative real part), or center (all eigenvalues purely imaginary).
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3. Consider the linear system (-2 Y dY dt (a) Determine the type of the equilibrium point at the origin. (b) Find the eigenvalues and eigenvectors and sketch the phase plane.
5. Solve the linear systems using eigenvalues and eigenvectors. dx = x+4z dt dy dt -= 2y dz -= 3x+y-3z dt
For the system of equations ()- dæ -3 -1 x. dt 1 1. Find the eigenvalues and eigenvectors of the matrix. 2. Find the coordinate transformation that will put the equation into canonical form and show that this works. 3. Write down the general solution.
Find the General Solution of the given systems. Make sure to indicate your Eigenvalues and Eigenvectors.
1b

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Consider the coupled linear system i = 3y – 4x ý = 5y – 6x 1. Write this system in the form dæ Ax. %3D dt 2. Find the eigenvalues and eigenvectors of A. What is the stability type of the origin? 3. Find the general solution and draw a phase portrait (remember to label everything!)