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 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 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.

The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system. x', = 4x, + 2x2 + 2x3, x'2 = - 6x, - 4x2 - 2x3, x'3 = 6x, + 6x2 + 4x3 What is the general solution in matrix form? x(t) =

The coefficient matrix has a zero eigenvalue. As a result, the pattern of trajectories is different from those in the examples in the text. For each system: Find the general solution of the given system of equations.

SOLVE STEP BY STEP IN DIGITAL FORMAT PLEASE Through the point P(-2,-7) tangents are drawn to the ellipse 2x² + y² + 2x-3y - 2 = 0; determine the equations of the tangents and the points of tangency +