1. a) Given the matrix A( 1²3 -¹1₂2).5ponding eigenvectors of A.b) Determine the general solution of the system of linear differential equations:Jx¹{Determine the eigenvalues and corres-= 2x1 + x₂5x1 -2x2x'2 =c) Determine the solution considering an initial condition: x(0)[2].thatmeans, r₁(0) = 2 and ₂(0) = 2. What is the behavior of this solution ast → ∞?=d) Determine the solution considering an initial condition: x(0) = [ 0-2 ],thatmeans, x₁(0) = 0.2 and x₂(0) = -1. What is the behavior of this solution ast → ∞?

As per our guidelines, we can answer only first three sub-parts. Kindly repost rest questions…

Answered: 2) 1. a) Given the matrix A = ponding…

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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Given the matrix A, below, the system below has a nontrivial solution corresponding to the eigenvalue 0.8 - 0.6 i. Solve the first equation for x, in terms of x1, and from that produce the eigenvector y = for matrix A. Show that this y is a complex multiple of the vector -6- i which is a basis for the eigenspace corresponding to A = 0.8 - 0.6 i. 37 V4 - 2.8 - 0.6 A= (-3.6 + 0.6 i )x, - 0.6x, = 0 22.2 22.2x, + (3.6 + 0.6 i )x2 = 0 4.4 Solve the first equation, (-3.6 +0.6 i )x, - 0.6x, = 0 for x, in terms of x,
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.
Suppose A is a 2 × 2 real matrix with an eigenvalue λ = 3 + 5i and corresponding eigenvector i * = (-²+1). v= Determine a fundamental set (i.e., linearly independent set) of solutions for ÿ' = Aỹ, where the fundamental set consists entirely of real solutions. Enter your solutions below. Use t as the independent variable in your answers. (t) = 2(t) = → - →
1. Find the eigenvalues and the corresponding eigenvectors of the matrix A in the following system of linear equations 3x1 +x2 = Àx1 -5x1 – 3x2 = Ax2 %3D
If A is an (n x n)-matrix of real constants that has a complex eigenvalue and eigenvector v, then the real and imaginary parts of w(t) = elty are linearly independent real-valued solutions of x' =Ax: x₁(t) = Re(w(t)) and X₂(t) = Im(w(t)), The matrix in the following system has complex eigenvalues; use the above theorem to find the general (real-valued) solution. 0 30 -3 0 0 X 005 x(t) = = x' = Find the particular solution given the initial conditions. x(t) = = x(0) = 123
Q3. (a) If Ax = 2 x, determine the eigenvalues and the corresponding eigenvectors for A= G ) 2 3. Hence, write down the associated modal matrix P and diagonal matrix D, and use these values to solve the following system differential equations: *1 = X1 + 4 x2 X2 = 2 x1 + 3 x2 Given that when t = 0, x1 = 0 and x2 = 2. (b) Use the 4th order Runge Kutta method to solve the differential equation: dy e* y dx for values of x 0 (0.2) 0.4 given that y 1 when x 0. %3D Give your answers correct to 5 decimal places. Obtain the analytical solution of the differential equation and compare the analytical solution when x = 0.4 with the values obtained using Runge Kutta,
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