You have the following initial value problem:xi= 6x2X₁ (c)=33x ₂² = -6-x₁ + 12x₂ 3x²₂ (0)=59Determine eigenvalue, eigenvector as well as a generalizedeigenvector for the coefficient matrix of the system shownabove.λ =v₁ =Figure out the most real valued general solution to . c1 and c2should be used for any arbitrary constants.x₁ (t) =x₂ (t) =You must solve the original initial value problemX₁ (t) =x₂ (t)=

The given system of equation is The system of equation can be written in matrix form as:…

Answered: You have the following initial value…

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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You are given the following matrix: [1 1][4 1] You find the eigenvectors to be: v1 and v2 [-1] [1][2] [2] Express the vector v = [4 4]T in the form av1+ βv2 where v1 and v2 are the eigenvectors above
Suppose that the matrix A has the following eigenvalues and eigenvectors:
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Consider the following interpolating points for the function f(x) = x² – 2x: Xi 01 23 f(xi) 0 -1 03 (a) Use monomial basis to find P₁(x) that interpolates f(x) at x; for i = 0,1. Con- struct the Vandermonde matrix and then use the backslash \ in matlab to solve the linear system. (b) Repeat (a) to find P₂(x) that interpolates f(x) at x; for i = 0, 1, 2. (c) Repeat (a) to find P3(x) that interpolates f(x) at x, for i = 0, 1, 2, 3. (d) Compare P₂(x), P3(x) and f(x). What do you see? Can you explain it?
You may use a calculator to perform basic calculations on these problems: (just label when using calculator) Row reduction Matrix multiplication Matrix inverses Finding eigenvalues and eigenvectors
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. x', = - 2x, + 8x2. x'2 = 12x, - 6x2 What is the general solution in matrix form? x(t) =O
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 = 9x₁ - 10x₂, x 2 = 8x₁ + x₂ What is the general solution in matrix form? x(t) = F More e
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
Solve the following matrix system by using variation of parameters: - () () dx 5 1 x + 5e-t dt 4 -5e-t Note that the eigenvalues of the matrix are -1, –6 with respective eigenvectors () | 4.
Everything we've done with systems of 2 linear, constant co- efficient, homogeneous differential equations works for larger systems as well. Ask wolframalpha.com for the eigenvalues and eigenvectors of the following system of three differential equations, and use the result to write the general solution: -2 -4 2 *m-Biss X' (t) -2 1 2 X(t) 4 2 5 =

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