6. The nonlinear systemy₁ = -y₁ - y2,y₂ = y₁ + y₂,has steady states at (0,0), (1, -1), and (-1,1). Classify each steady state as a source (alleigenvalues 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 (alleigenvalues purely imaginary).

Answered: 6. The nonlinear system y₁ = -9₁-92, y₂…

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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Express the given system of higher-order differential equations as a matrix system in normal form. x"' + 3x+6y=0 y" - 5x=0 Which of the following sets of definitions allows the given system to be written as an equivalent system in normal form using only the new variables? O A. X₁ X₁ X₂=X", X₂ =Y₁X₂ =y" OB. X₁ X¹, X₂=X", X₂ = Y', X₂=y" OC. X₁ = x¹, x₂ = y' O D. X₁ =X, X₂=X', X3 = Y, X4 = Y' Write the system of equations using matrix notation. Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.) O A. co Xq' X₂ m A 11 II X₁ X₂ X3 X4
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Consider the system of differential equations For this system, the smaller eigenvalue is (-39/10) [Note-- you may want to view a phase plane plot (right click to open in a new window).] If y'= Ay is a differential equation, how would the solution curves behave? • All of the solutions curves would converge towards 0. (Stable node) All of the solution curves would run away from 0. (Unstable node) and the larger eigenvalue is The solution curves would race towards zero and then veer away towards infinity. (Saddle) The solution curves converge to different points. The solution to the above differential equation with initial values (0) = 4, y(0) = 3 is (t)= (49/12)*e^((-9/10)*t)-(1/12)*e^((-19/10)*t) y(t) = (49/12)*e^((-9/10)*t)+(5/12)*e^((-19/10)*t) da dt dy dt = -1.4x +0.75y, = 1.66667 3.4y. (-9/10)
The behaviour of an unforced mechanical system is governed by the differential equation 5 2 -1 0 x(1) = 3 6-9 x(t), x(0) = 1 1 1 1 0 (a) Show that the eigenvalues of the system matrix are 6, 3, 3 and that there is only one linearly independent eigenvector corresponding to the eigenvalue 3. Obtain the eigenvectors corresponding to the eigenvalues 6 and 3 and a further generalized eigenvector for the eigenvalue 3. (b) Write down a generalized modal matrix M and confirm that AM = MJ for an appropriate Jordan matrix J. x(t) = Me M-¹x(0) obtain the solution to the given differential equation. (c) Using the result
A system of ordinary differential equations has 2 x 2 state matrix with eigenvalues and corresponding eigenvectors (eigenpairs) A1 = 5, v1 = and A2 = -6, Based on the given eigenpairs write down the general solution to the associated system of differential equations. The solution must be written as a |¤1(t) single vector like a =
The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system. x₁ = 7x₁ + x₂ + x3, x2 = - 13x₁ - 7x₂ -X3, X'3 = 13x₁ + 13x2 + 7x3 * A What is the general solution in matrix form? x(t) = Jiry
Consider the system of differential equations = (-3/2 3/2)x+ (-¹1) ¹ x' = -3/2 -1 where two eigenvalues of 2 3/2 -3/2 -1 are r = 1/2 and r₂ = 1/2, and the corresponding eigenvector is v= -(2/3). Let x = Ty where T is a transformation matrix. Find two first order differential equations, eigenvector is u=== and = -/+nte, n =+², ₂ -+-+ M and the generalized
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Suppose the eigenvalues associated with a 2 by 2 matrix of a system of linear ODE's are complex conjugates. (a) Use Euler's formula to rewrite the solution in the form x(t) = u(t) + iv(t). (b) Show that x(t) = u(t) + v(t) is also a solution of the system.
Suppose a system of first-order linear differential equations has eigenvalues A₁ = -3, A2=-2.3. The equilibrium point (origin) can be classified as: O Flux Point O Saddle Point O Origin Point O Stable Node (Sink) O Unstable Node (Source)
Given the system x' = Ax = -15 8 -16 4 -10 5 7 -1 -7 2 8 -5 -1 1 -2-2- x, find its eigenvalue(s) and eigenvectors. Determine the four eigenvalues of A. Determine all of the ordinary eigenvectors of A that you can. For each unique eigenvalue, what is its multiplicity k, and how many defects d does it have? Determine any missing eigenvectors. You should use the Excel "=MMULT(matrix, matrix)" (the form of the Excel function) for your work. Otherwise, simple arithmetic mistakes are too likely. Determine the four linearly independent solution vectors, designating them x₁(t), x₂(t), x3(t), and X4(t). Write the general solution vector x(t) as a linear combination of the four solution vectors from the question immediately above, using coefficients c₁ through c4, respectively, but leaving them as unknowns. You are encouraged to write this solution row-by-row, such as x₁(t) = ..., X₂(t) = ..., and so on, for rows 1, 2, and so on, respectively. Note that these x's are functions, not vectors. Form…

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