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,
With two independent eigenvectors?
Consider a four-state system whose Hamiltonian is given by Vo +A 8Vo H 3Vo -2) 7Vo -2) where Vo is a real-valued constant and A is a real-valued parameter. (c) Now decompose this Hamiltonian into an unperturbed Hamiltonian, Ho, and a perturbing piece, H'. Solve the unperturbed system, that is, find the (four) eigenvalues and (four) eigenvectors of your unperturbed Hamiltonion, Ho. (d) Using nondegenerate perturbation theory, find the first- and second-order corrections to the unperturbed energies in the presence of the perturbing Hamiltonian, H'. (e) Solve the system exactly, by diagonalising the full Hamiltonian, H, to find the exact eigen- values of the full Hamiltonian, H. (f) Expand your solutions to part (e) to the appropriate order in A and compare these results to your results from nondegenerate perturbation theory in part (d). What condition must you impose on A to ensure that the leading energy correction from perturbation theory is a good approximation to the exact result?
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,
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.
f) Use your eigenvalues and eigenvectors to write the general solution, in matrix form, to your linear system from part (a). (Your answer should have two arbitrary constants). g) Apply your initial conditions from part (a) to solve for the arbitrary constants. Please answer a-h with clear steps thanks h) Take your solution out of matrix form and rewrite it in the form Q1= f(t), Q2 = g(t)
1b
can someone explain this for me, please ?

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