Consider the following system.λ =dxdtdydtFind the eigenvalues of the coefficient matrix A(t). (Enter your answers as a comma-separated list.)K == 6x + 5y= -2x + 8yFind an eigenvector corresponding to the eigenvalue with positive imaginary part.=Find the general solution of the given system.(x(t), y(t))

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Answered: Consider the following system. dx dt dy…

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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Consider the linear system A₁ = a. Find the eigenvalues and eigenvectors for the coefficient matrix. = 0 , V1 = 1(t) = = -2t -2 3t v=13&v 3 y. " and X2 = 1 b. For each eigenpair in the previous part, form a solution of ÿ' = Aÿ. Use t as the independent variable in your answers. and y₂(t) = -1t , V₂ 2t || -1 2 c. Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solutions? Yes, it is a fundamental set +
( 3). Consider the system of equations x' = Ax where A = The eigenvalues for the matrix A are d1 = 2, A2 = 3. Find the corresponding eigenvectors vị and v2.
Consider the second-order equation d 2y/dt 2 + p dy/dt +qy = 0 a. If p=q=1, compute the eigenvalues and eigenvectors of the coefficient matrix. What would be the long-term behavior of the solutions in this case? b. If p=0 and q=2 determine the general solution.
Suppose that the matrix A has the following eigenvalues and eigenvectors:
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Linear algebra: please correctly and handwritten
Consider the linear system dx₁ dt dx₂ dt = 4x1 - 3x₂ = 3x1 + 4x2. 1. Show that upon conversion to the form X'= AX, the eigenvalues of A are complex. 2. Find the general solution of the system.
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. For the system below, find the general solution, sketch the trajectories, being careful to include the eigenvector directions, and classify the type of fixed point: x = x, ÿ y = 2x - 5y.
) Consider the linear system X₁ = Y a. Find the eigenvalues and eigenvectors for the coefficient matrix. 18 v1 = b. Find the real-valued solution to the initial value problem {{{ = -3 -2 5 -3] Use t as the independent variable in your answers. y₁ (t) Y₂ (t) - 3y1 - 2y2, 5y1 + 3y2, y." , and X₂ = V2 y₁ (0) = 9, Y₂ (0) = -10. ||
Consider the system of equations vec(x′) = Ax, where A is a 2×2 matrix. If λ=2+i is an eigenvalue of A with corresponding eigenvector vec{v} = (1, −1−i), express the solution to the system of equations in terms of real-valued functions.
Consider the linear system -3 = 5 a. Find the eigenvalues and eigenvectors for the coefficient matrix. λ₁ = v₁ = = b. Find the real-valued solution to the initial value problem Use t as the independent variable in your answers. y₁ (t) = y2(t) = and ₂ = = = -3y1 - 2y2, y₁ (0) = 0, = 5y1 + 3y2, y2 (0) = 5.

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