In each of Problems
(a) Determine all critical points of the given system of equations.
(b) Find the corresponding linear system near each critical point.
(c) Find the eigenvalues of each linear system. What conclusions can you then draw about the nonlinear system?
(d) Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system.
(e) Draw a sketch of, or describe in words, the basin of attraction of each asymptotically stable critical point.
Want to see the full answer?
Check out a sample textbook solutionChapter 7 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
Additional Math Textbook Solutions
Finite Mathematics & Its Applications (12th Edition)
Thinking Mathematically (7th Edition)
Introductory Mathematics for Engineering Applications
Calculus Volume 1
Thinking Mathematically (6th Edition)
The Heart of Mathematics: An Invitation to Effective Thinking
- Please help with my homeworkarrow_forwardIn each of Problems 5 and 6 the coefficient matrix has a zero eigenvalue. As a result, the pattern of trajectories is different from those in the examples in the text. For each system: Ga. Draw a direction field. b. Find the general solution of the given system of equations. G c. Draw a few of the trajectories. 4 -3 8 -6 5. x' = Xarrow_forward1. For the system below, find the general solution, sketch the trajectories, being careful to include the eigenvector directions, and classify the type of fixed point: x = x, ÿ y = 2x - 5y.arrow_forward
- Need Handwritten SOLUTION.Need Solution in 30 Minutes .Solve Q2 only .Thankyou!!arrow_forwardQ1. (a) (b) Derive the row-echelon form for the following set of equations and state the solution (if it exists). 4x-2y-z=3 -x + 3y + 5z = 9 x + 2y +4.5z = 10.5 5x + 5y + 13z = 33 Obtain the eigenvalues and eigenvectors of the matrix 2 - 8 - 4 6arrow_forwardConsider the linear system 3 *' = 27 a. Find the eigenvalues and eigenvectors for the coefficient matrix. help (numbers) help (matrices) A1 = , 01 = and help (numbers) help (matrices) 12 = , v2 = Find the real-valued solution to the initial value problem x = -3x1 – 2x2, x = 5x1+3x2, x1(0) = -4, x2(0) = 10. Use t as the independent variable in your answers. help (formulas) T2(t) help (formulas) OK Learn more Cookies help us deliver our services. By using our services, you agree to our use of cookies.arrow_forward
- Please show step-by-step solution and do not skip steps. Explain your entire process in great detail. Explain how you reached the answer you did.arrow_forward(a) convert the equation to a first-order, linear system; (b) compute the eigenvalues and eigenvectors of the system; (e) for each eigenvalue, pick an associated eigenvector V, and determine the solution Y() to the system; and (d) compare the results of your calculations in part (c) with the results that you ob- tained when you used the guess-and-test method of Section 2.3. d²y d12 dy +5+6y=0 dtarrow_forwardFor 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 - 6yarrow_forward
- Apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, use graphing calculator to construct a typical solution curves for the given system. x = 9x1 + 5x2 х 3 — 6х1 — 2х2 X1 (0) = 1, x2(0) = 0arrow_forwardFor the following systems, the origin is the equilibrium point. dx a) Write each system in matrix form = Ax. dt 5. 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. 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. f) If A has real eigenvalues, then determine the eigenvectors and use diagonalization to solve the system. (See examples in Section 7.4) dx dt dy dt = -3x + 4y = 2x - 5yarrow_forwardI got this question rejected as marked part of assignment but it’s not. IT’s question for practice before exam and i don’t know how to do it. Help !arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning