Consider the following system.dxdt= 6x + y,dydt= -2x + 4yFind the eigenvalues of the coefficient matrix A(t). (Enter your answers as a comma-separatedλ = 5+1,5-iFind an eigenvector corresponding to the eigenvalue with positive imaginary part.K =<1,-1+i>Find the general solution of the given system.5t(x(t), y(t)) =

System of differential equation and it's solution

Answered: Consider the following system. dx dt…

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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(1 point) Consider the linear system -3 -2 y. 3 a. Find the eigenvalues and eigenvectors for the coefficient matrix. and X2 b. Find the real-valued solution to the initial value problem — Зул — 2у2, ) = 4, 5y1 + 3y2, y2 (0) = -10. Use t as the independent variable in your answers. Y1 (t) = Y2(t) = I||| 15
Consider the following system. dx dy = -2x + 5y dt dt Find the eigenvalues of the coefficient matrix A(t). (Enter your answers as a comma-separated list.) A-6-1,6+i = 7x + y, Find an eigenvector corresponding to the eigenvalue with positive imaginary part. K= Find general solution of the given system. (x(t), y(t)) - et (cos(t) (1, -2)-sin(t) (1,0)), et (cos(r) (1,0) + sin() (1, -2)) Need Help? Read It x
Find the matrix A of the quadratic form associated with the equation. 3x2 - 8xy - 3y² + 15 = 0 A Find the eigenvalues of A. (Enter your answers as a comma-separated list.) Find an orthogonal matrix P such that P'AP is diagonal. (Enter the matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list.) P =
4) Solve the following linear system or indicate the nonexistence of solution. X1- 2x2 +5x3- 3x4 0 -3x1 + 6x2 + x3 + x4 0 2x1 - 4x2 + 3x3 – x4 = 3 5) Find the eigenvalues and eigenvectors of the following matrix. (0 € R) [cos 0 Isin 0 - cos 0. sin 0
Suppose A is a 2 × 2 real matrix with an eigenvalue A = 1+ 4i and corresponding eigenvector -1+i Determine a fundamental set (i.e., linearly independent set) of solutions for y' = Aj, where the fundamental set consists entirely of real solutions. Enter your solutions below. Use t as the independent variable in your answers. 1 ÿ1(t) ý2(t) =
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.
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
) 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. ||
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
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.
Suppose the eigenvalues associated with a 2 by 2 matrix of a system of linear ODE's are complex conjugates. (a) Use Euler's formula to rewrite the solution in the form x(t) = u(t) + iv(t). (b) Show that x(t) = u(t) + v(t) is also a solution of the system.
Xn Let vn be the solution to the discrete dynamical system: Yn 6xn – 13yn Oxn – 7yn Xn+1 Yn+1 [:] -3 If vo find vn- -1 Put the eigenvalues in ascending order when you enter xn, Yn below. Xn = Yn = )"+ )" ||

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