11 Nonlinear systemFind all three equilibrium points of the following nonlinear system Then classify (asstable/unstable) any two of them. No need to calculate the eigenvectors.r = 2²¹-ry²y = ry+3y

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Answered: Find all three equilibrium points of…

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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2. For the system of equations: dx = aj|x + a12y, dt dy a21x + az2y. dt First, review lecture notes to see how we eliminate one variable to arrive at a single second-order equation. Then find the characteristic equation and solve the eigenvalues of the equation. Once x(t) is found, y(t) can be found by setting 1 dx - a11x) y(t) = (a12 # 0) a12 dt By using the above method, find solutions for the following systems: (a) dx = -4x + y, dt dy = 3x. dt (b) dx = 2x – 3y, dt %3! dy = x- 2y. dt
X₁ = Consider the following system: Find the eigenvalues and eigenvectors of the coefficient matrix A. V₁ |x₁ = I2 = dt -X and X2₂ = Sketch a phase portrait. Solve the system given initial conditions ₁ (0) = 0, x₂(0) = -17 8 -52 23 -1. V₂ Note: When Webwork checks whether eigenvectors and eigenvalues are correct, ALL eigenvalues and eigenvectors are evaluated at once. So individual eigenvectors and eigenvalues won't be marked correct unless all the eigenpairs are correct. I =
Find all the equilibrium points for the system below. At each equilibrium point, linearize the system and classify whether the equilibrium is small signal stable or unstable by computing the eigen values at each equilibrium point dx, -2x1+x1*X2 dt dx 2 = -2x,+x;*X2 dt
Consider the second-order equation d 2y/dt 2 + p dy/dt +qy = 0 a. If p=q=1, compute the eigenvalues and eigenvectors of the coefficient matrix. What would be the long-term behavior of the solutions in this case? b. If p=0 and q=2 determine the general solution.
For the following systems, the origin is the equilibrium point. 3. a) Write each system in matrix form b) Determine the eigenvalues of A. c) State whether the origin is a stable or unstable equilibrium. d) State whether the origin is a node, saddle point, spiral point, or center. dx dt e) State the equations of the straight-line trajectories and tell whether they are going towards or away from the origin. If none exist, state so. dx dt dy dt = Ax. f) If A has real eigenvalues, then determine the eigenvectors and use diagonalization to solve the system. (See examples in Section 7.4) = 4x - 13y = 2x - 6y
) Consider the linear system X₁ = Y a. Find the eigenvalues and eigenvectors for the coefficient matrix. 18 v1 = b. Find the real-valued solution to the initial value problem {{{ = -3 -2 5 -3] Use t as the independent variable in your answers. y₁ (t) Y₂ (t) - 3y1 - 2y2, 5y1 + 3y2, y." , and X₂ = V2 y₁ (0) = 9, Y₂ (0) = -10. ||
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dy dy +3 dr? 6. Consider the second-order equation - 10y = 0. di (a) Convert the equation to a first-order, linear system. (b) Compute the eigenvalues and eigenvectors of the system. (c) For each eigenvalue, pick an associated eigenvector V, and determine the solution Y(t) to the system. (d) Compare the results of your calculation with the results that you obtained when you used the guess-and-test method in Homework 6, question 4.

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Find all three equilibrium points of the following nonlinear system. Then classify (as stable/unstable) any two of them. No need to calculate the eigenvectors. x = 2²³²-ry² y = xy + 3y