Multiple Choice) Consider the phase portrait below for the system x Ax,> 0.4-2We 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 givenWrite 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 given2.

Answered: Image /qna-images/answer/d5691ee8-359d-44d5-a7c4-3c8ee5aa3d34.jpg

Answered: Multiple Choice) Consider the phase…

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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Apply the eigenvalue method to find a general solution of the given system. Use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. x₁ = 2x₁ + 3x2 x'2 = 3x₁ + 2x2 *** The eigenvalue(s) of the constant coefficient matrix is (are) (Use a comma to separate answers as needed. Simplify your answer. Type an exact answer, using radicals and i as needed.)
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)
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Use (a) to solve the system x = (: :) . Х. -8 7 NB: Real solutions are required. Hint: Recall that if the roots of C(A) = 0 occur in complex conjugate parts, then any one of the two eigenvalues yields two linearly independent solutions. The second will yield solutions which are identical (up to a constant) to the solutions already found. a) is as follows Show from first principles, i.e., by using the definition of linear independence, that if x+iy, y # 0 is an eigenvalue of a real matrix A with associated eigenvector v = u + iw, then the two real solutions Y(t) = et (u cos bt – w sin bt) and Z(t) = eat (u sin bt + w cos bt) are linearly independent solutions of X = AX.
1 Let P = -2 - Find p P. Deduce P, if it exists, from the result of -1 p' P. You can also calculate P 1 using the traditional approach but this will be slower. 2. Let A be a 3 x 3 matrix with the following eigen-pairs: 1 (1, ):(1, 0 ):G -2 ).
Solve only (c) part
The coefficient matrix A of the 4x4 system below has eigenvalues A₁-9, A₂-6, Ag 9, and 4 14. Find the particular solution of this system that satisfies the initial conditions x₁ (0) = 5, x₂ (0) = 5, x₁ (0) = 5, x4 (0) = 5. x₁ = 2x₁ + x₂ + xy + 8x4 x2 = x₁ + 2x₂ + 11xg+X4 X3 X₁ +11x₂ + 2xy +Xg x48x₁ + x₂ + xy + 2x₁ *(t) =
lloa [Ur ses SEC::116 Question 9 Find the general solution to the system x' = Ax where A is the given matrix. -2 -1 1 A= -1 -9 3 -1 -2 -4 Hint: -6 is an eigenvalue.
Show your work
Find the general solution of the following inhomogeneous systems. Use either eigenvector decomposi- tion or undetermined coefficients method. These problems can be solved by hand, but you can also use technology to assist you at any stage. When using technology always report on what function you use. **BHRAIBHS] -2 5 5 -2 (a) = (b) + = 2 -1 **[G]HBKH[**] 3 -2 3e + -2t cos 3t

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