Consider a system 7'a) Find the eigenvalues of A.b) What is the type of the Phase-Plane portrait ? Is the origin attractingor repelling, or neither attracting nor repelling ? Is the origin stable ?c) Find eigenvectors. Then use eigenvalues and eigenvectors to write a gen-A and answer the following questions.eral solution.d) Use any method to find the matrix exponential etA.e) Find the solution of the given initial value problem.f) In the phase plane sketch eigenlines and sketch the solution curve 7(t), –∞ <t < o, of the IVP. DO NOT sketch other solution curves.Clearly show two parts of your curve : one corresponding to t > 0,corresponding to t < 0. Use arrows to show the direction of increasing t onthe curve and on the eigenlines. Mark the point (0). Evaluate the tangentvector to the solution curve at (0) and sketch it.another()T , 7(0) =31

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Consider a system a' = Ax and answer the following questions. a) Find the eigenvalues of A. b) What is the type of the Phase-Plane portrait ? Is the origin attracting or repelling, or neither attracting nor repelling ? Is the origin stable ? c) Find eigenvectors. Then use eigenvalues and eigenvectors to write a gen- eral solution.

Consider a system a' = Ax and answer the following questions. a) Find the eigenvalues of A. b) What is the type of the Phase-Plane portrait ? Is the origin attracting or repelling, or neither attracting nor repelling ? Is the origin stable ? c) Find eigenvectors. Then use eigenvalues and eigenvectors to write a gen- eral solution.

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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Question

Consider a system 7'
a) Find the eigenvalues of A.
b) What is the type of the Phase-Plane portrait ? Is the origin attracting
or repelling, or neither attracting nor repelling ? Is the origin stable ?
c) Find eigenvectors. Then use eigenvalues and eigenvectors to write a gen-
A and answer the following questions.
eral solution.
d) Use any method to find the matrix exponential etA.
e) Find the solution of the given initial value problem.
f) In the phase plane sketch eigenlines and sketch the solution curve 7(t), –∞ <
t < o, of the IVP. DO NOT sketch other solution curves.
Clearly show two parts of your curve : one corresponding to t > 0,
corresponding to t < 0. Use arrows to show the direction of increasing t on
the curve and on the eigenlines. Mark the point (0). Evaluate the tangent
vector to the solution curve at (0) and sketch it.
another
()
T , 7(0) =
3
1

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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?
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The eigenvalues in this problem are all nonnegative. First determine whether A=0 is an eigenvalue; then find the positive eigenvalues and associated eigenfunctions. 7y +Ay=0; y(-x)=0, y(x)=0 Is A = 0 an eigenvalue of this problem? Select the correct answer below and, if necessary, fill in the corresponding answer box to complete your choice. OA. Yes, A-0 is an eigenvalue with the corresponding eigenfunction yo(x)= OB. No, when A=0 the only solution to the given equation is the trivial solution. OC. No, when A=0 there are an infinite number of nontrivial solutions to the give equation. What are the eigenvalues and eigenfunctions of the problem for n=1. 2. 3...? Select the correct answer below and fill in the corresponding answer box to complete your choice OA. The eigenvalues are À = OB. The eigenvalues are À = The eigenfunctions are yn(x)= The eigenfunctions are y(x) = regardless of whether n is even or odd when n is even and y(x) when n is odd.