Suppose that the differential equation x' = Ax is such that A is a 2 × 2 matrix with (1) eigenvalues A1 = 5 and A2 = 3 and corresponding eigenvectors vị = and v2 = the solution that satisfies the initial condition x(0) : Explain what happens to the trajectory of this solution as t → 0. Circle one of the items below. i. becomes parallel to the vector vị = (-). ii. becomes parallel to the vector v2 = iii. tends towards the point (0,0). iv. Spirals away from the origin v. none of the above

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Suppose that a 2 x 2 matrix A with real entries has an eigenvalue X = 1 + 3i with corresponding 4 i [12] eigenvector 1 + 2i Which of the following is NOT a solution of the linear system y' = Ay? y(t) = et y(t) = et y(t) = et y(t) = et 4 cos(3t) sin(3t) [cos(3t) + 2 sin(3t). 4 cos(3t) + sin(3t) cos(3t) 2 sin (3t) - cos(3t) + 4 sin(3t) 2 cos(3t) + sin(3t) cos (3t) - 4 sin (3t) -2 cos(3t) - sin(3t)

Transform the given system of differential equations into an equivalent system of first-order differential equations. x' - 3x' - 5x+2y=0 y'' +6y' - 6x-3y = sin t Let x₁ = x, X₂ = X', Y₁ = y, and y₂ = y'. Complete the system below. X₁' X₂ X Y2 II 11 11 11