Sketch several solution curves in the phase plane of the system of differential equations dx/dt = Axusing the given eigenvalues and eigenvectors of A.A₁ = 2, A₂ = -2;*-8 »-8V₁ ==

The general solution of the given system of DE using the given data is

Answered: Sketch several solution curves in the…

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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1 0 2. Consider the identity matrix / = 1 a. Come up with three linearly independent eigenvectors of I. b. Orthonormalize your three eigenvectors.
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Consider the matrix 11 3 3 19 The eigenvector corresponding to the eigenvalue 20 can be written in the form [1 y² Enter the value of y in the box below, rounded to the nearest integer. Answer:
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2. 0 1 0 -89 0 0 1 -12 -19 -8] Let A = (a) Find all the eigenvalues and eigenvectors of A. (b) Suppose x, y and z are functions of t. Find the general solution of the following system of DES. x' = 1 y' = y 2 -12x 19y - 8z (c) By part (b), find the particular solution of the following IVP. d³x d²x +8. +19- dt³ dt² dx + 12x = 0, x(0) = 1, x(0) = 1, x'(0) = 1, x"(0) = 1. dt
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With two independent eigenvectors?
2. (1) Compute all eigenvalues and eigenvectors of the Cartan matrix 2 -1 C = -1 2 -1 -1 2 (2) Diagonalize C (i.e. find Q such that Q-'CQ is diagonal). (3) Is C positive definite?
35. (a) Show that the eigenvalues of the 2 X 2 matrix a b A = c d] с а are the solutions of the quadratic equation 12 – tr(A)A + det A = 0, where tr(A) is the trace of A. (See page 162.) (b) Show that the eigenvalues of the matrix A in part (a) are A = {(a + d ± V(a – d)? + 4bc) (c) Show that the trace and determinant of the matrix A in part (a) are given by tr(A) - λι + λ and det A = λιλε where A, and A2 are the eigenvalues of A. 36. Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has (a) two distinct real eigenvalues, (b) one real eigenvalue, and (c) no real eigenvalues.
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Sketch several solution curves in the phase plane of the system of using the given eigenvalues and eigenvectors of A. [3] [1]