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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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(4.2) Use the eigenvalue-eigenvectors to solve the initial value problem X 200 −1 -1 1 0 X, -232 - 1 X(0) = 2 -2
Apply the eigenvalue method to solve the initial value problem:
x1 = x2 – 4x + 4x3 %3D 4 x½ = -3x2 - V3x, + V3x1 +X3 x3 = 7x2 - 2x1 Find the eigenvalues and eigenvectors by using the above equations.
11) Find the values of such that the equations have non-trivial solutions. What are the eigenvalues and eigenvectors? (6-2) x + 3y = 0 3x - (2+2)y = 0
A = [10 5 -1 Use four iterations of the power method to approximate an eigenvector and eigenvalue of -12 -6 -5 -4 using To= 0 3 [!] 5.
5) Find the values of that such that the given equations have non-trivial solutions. What are the eigenvalues and eigenvectors? (1-2) x + 2y = 0 2x + (4-2)y=0
Approximate the eigenvalues of
Exercise: Let A be an invertible n x n matrix. Suppose that u ER" is an eigenvector of A with corresponding eigenvalue λ = 2 and v ER" is an eigenvector of A with corresponding eigenvalue = -7. Fill in the following blanks. (Enteras a/b, where applicable. Work out any powers before inputting your answer.) 1. Au = 3. Asu = 5. A u = u u u 2. Av= 4. A³v = 6. A-²y = V Example: Suppose that A is an invertible n x n matrix and that x ER" is an eigenvector of A with corresponding eigenvalue λ = -3. Since x is an eigenvector of A with corresponding eigenvalue λ = -3 we have that x = 0 and Ax = X. (a) Show that x is also an eigenvector of B = 2A²-6A-¹ + 51 and find the corresponding eigenvalue (of B).
[3 2] ? Why or why not? 3 8 2. Is A = 2 an eigenvalue of
2 2 -6 8. The general solution of x' = x is: (Hint: 6 is an eigenvalue.) 2 -1 -3 -2 - 1 -2 3 1 (a) x = Cieốt + Cze-2t + C3 3 (b) x = Ce6t 1 + Cze-2t 2 (c) x = Ce6t + Cze-2t + Cze-2t 2 2 2 3 (d) x = Cie6t + C3 te-2t 2 1 + Cze-2t -2 (e) None of the above.
The matrix v₁ = 1 x₁ (t) = -13 12] -8 7 x₂ (t) = has eigenvalues A₁ = -1 and ₂ = -5. Find eigenvectors corresponding to these eigenvalues. satisfying the initial conditions 1 2 Find the solution to the linear system of differential equations and 7₂ x₁ (0)] _x₂ (0) = [3] help (formulas) help (formulas) = x 3 x = - 13 -8 12 7 X1 X help (matrices)
LA-(2-2) = Let 0 1. Calculate the eigenvalues of A. 2. And if possible, construct the diagonal factorization A = PDP-¹ ( You don't need to calculate P-¹).

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(1.2) Use the eigenvalue-eigenvectors method to solve the initial value problem -2 1 1 -2 X = 0 1 -3 X, X(0) = 2 00 -2 -1