Question

For the system:

a. Write the general solution

b. State if the origin is a spiral sink, source, or center and explain why.

c. Write the natural period and natural frequency of the solutions.

d. Do the solutions go clockwise or anticlockwise around the origin? Explain your reasoning.

e. Write the particular solution that corresponds to

Please write clearly and explain your steps. Thank you! Do not just copy someone else's answer.

Answer 1

Given data dy/dt = [2 - 1 2 2] y y (t) = [x(t) y(t)] Let y = (x(t) y(t)Therefore = [2 - 1 2 2]y Rightarrow x^1 = 2x + 2y y^1 = -x + 2y---1) Therefore zy = x^1 - 2x----2) Therefore x^1 = 2x^1 + 2y^1 = 2x^1 - 2x + 4y = 2x^1 - 2x + 2x^1 - 4x Thus x^11 - 4x^1 - 6x = 0-----3) Characteristic equation of 3 is lambda^2 - 4 lambda - 6 = 0 Therefore lambda = 4 notequalto squareroot - 8/2 = 2 notequalto i squareroot 2 Therefore general solution 3 is x(t) = c_1 + 2 sin (t squareroot 2) --4) y(t) = 1/2 x^1 - x = 1/2 (squareroot 2 c_2 - 2c_2) e62t cos (t squareroot 2) + 1/2 (2c_2 squareroot 2c_1) e^2t sin (t squareroot 2)----5) \r\n general solution x(t) = c_1 e^2t cos (t squareroot 2) + c_2 2t (t squareroot 2) yet = 1/2 (squareroot 2.2)c_2 e^2t cos (t squareroot 2)-----6) It a flower which looks + 1/2 (2c_2 - squareroot 2c_1) e^2t sin (t squareroot 2)----7 The tag1)