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

4. Solve given that x' = Ax + A has eigenvalues -2 and -6. -3 3 1 5

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Suppose that the matrix A has the following eigenvalues and eigenvectors:

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