2. Consider the linear system x' = Aπ with = A-(G3). α where a -1. (a) Determine the eigenvalues in terms of a. (b) Determine the values of a, for which all solutions tend to zero as t→ ∞. (c) Suppose that a = -4, and *(0) = (70). For what values of xo and yo, is the solution (t) 0 as t→ ∞?

5c

For the system of differential equations, = A₁, A₂ = U₁ = U2 = y' a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) = [16 b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. = -16 21 C1 c) Find the eigenvectors assuming u₁ is the eigenvector associated with the smaller eigenvalue X₁ and u2 is the eigenvector associated with the larger eigenvalue X₂ . Enter the eigenvectors as a matrix with an appropriate size. 14 19 y d) Determine a general solution to the system. Enter your answer in the format y(t) = c₁f₁(t)ví + c₂f₂(t)v₂ . y(t) = + C₂

Please do number 4

Assume that you are solving the linear systems of differential equations given by i' = Ax and got the sets of eigenvalues given below. Match each set of eigenvalues with the appropriate description of the origin (which is an equilibrium solution). A). A = -3.2, 13 B). A = 2 + 3i C). A = ±8i D). A = -4, -9 E). A = 2, 18 F). A = -3 + 10i origin is an asymptotically stable node origin is an unstable node origin is a saddle point origin is an asymptotically stable spiral point origin is an unstable spiral point origin is a center

Please do number 5

b) please

Using the attached theorms, Solve the following parts: a) dx/dt =7x-y , dy/dt=5x+3y  b)dx/dt =-x+3y , dy/dt=-3x+5y

Solve the system of differential equations. Find the lesser and greater of the two eigenvalues.

Find all solutions of the following system of linear equations: dx dt = 4x + 8y, dy dt Xg = C1 = -2x - 4y Hi, I have to ask this question for the third time because the answers given were incomplete. Here, eigenvalue 0 is a repeated eigenvalue. So, answer includes two eigenvectors of corresponding eigenvalue, 0. Could you also show me how can I find the second eigenvector of it? Answer key says that the answer is: [1] +0₂² [2¹] -0₂H] 1-2