2. The unforced oscillator DE Y" + 8y' +12y = 0 may be written as a first-order vector DE x = Ax where A = (-12 -3) 1) where x(t) = +) _ (*1(t)) _ ( y(t) (x2(t)) (y'(t)) t Cae6t / 1 (y'(t)) = cie-2t / - 2 ' 1-6) (a) Sketch the phase portrait of the vector DE in the X1X2-plane, showing exceptional solutions, isoclines, and asymptotic behaviour as t + $0. (b) Answer the following questions about the oscillator using your phase portrait: i. Is the motion overdamped, underdamped, or critically damped? ii. If the initial conditions are y(0) = 0 and y(0) = 1, then is the oscillator speeding up (accelerating) or slowing down (decelerating) as it crosses the equilibrium position for the first time for t > 0? Why? (If it never crosses the equilibrium position for t > 0, then explain briefly why.) 2. The unforced oscillator DE Y" + 8y' +12y = 0 may be written as a first-order vector DE x = Ax where A = (-12 -3) 1) where x(t) = +) _ (*1(t)) _ ( y(t) (x2(t)) (y'(t)) t Cae6t / 1 (y'(t)) = cie-2t / - 2 ' 1-6) (a) Sketch the phase portrait of the vector DE in the X1X2-plane, showing exceptional solutions, isoclines, and asymptotic behaviour as t + $0. (b) Answer the following questions about the oscillator using your phase portrait: i. Is the motion overdamped, underdamped, or critically damped? ii. If the initial conditions are y(0) = 0 and y(0) = 1, then is the oscillator speeding up (accelerating) or slowing down (decelerating) as it crosses the equilibrium position for the first time for t > 0? Why? (If it never crosses the equilibrium position for t > 0, then explain briefly why.)