Solve the following systems of equations using the matrix method. Find eigenvalues and eigenvectorsby hand (but you can use technology to check your answers)(a)y'x + 2y= 2x + y"(b)x₁ =x₂=3x1 - 5x2x1 + x2

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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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Solve the system of differential equations given in matrix form for the general solution. We must do this by calculating the eigenvalues and eigenvectors. Note: we get a repeated eigenvalue and must solve for the second eigenvector to get two linearly independent solutions for our general solution. Please explain how you calculate the second eigenvector
Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are 10 and -9. A = 1 -3 9 -3 93 9 3 1 Enter the matrices P and D below. (Use a comma to separate answers as needed. Type exact answers, using radicals as needed. Do not label the matrices.)
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Thank you for your response, the calculations are clear up to factoring using the quadratic equations. however, I do not understand the concept behind for which values of k does A have two distinct real eigenvalues and when is there no real eigenvalues from the response. Please explain. Thank you
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Solve the following systems of equations using the matrix method. Find eigenvalues and eigenvectors by hand (but you can use technology to check your answers) I' = (a) y' = x + 2y 2x + y = . (b) 3x15x2 = 1+22