5. Consider the following linear systems and their general solutions: dx dy (a) = 6x-2y, =...

Transcribed Image Text:

5. Consider the following linear systems and their general solutions: dx dy (a) = 6x-2y, = 4x+2y dt dt with general solution x(t) 34 (1) ) = ar ( y(t) cos(2t) + sin(2t) dx dy (b) 5x+2y, = 4x+3y dt dt 4t et +2 with general solution x(t) (360) = 0) (1 ) " + (-2) 1 dx dy (c) = 3x - Y, = 6x - 4y dt dt et x(t) y(t) with general solution 1 = 1 e- = ((1)) - 9 (6) + (1) e2t dx dy (d) = -2x - Y, = x dt dt (s sin(2t) - 2 sin(2t) - cos(2t) e4t with general solution x(t) ( ) ) = ( 11 ) y(t) For each of the linear systems e-t + 2 ( -+ + 1 ) et (i) Sketch the phase portrait near the critical point at the origin. (ii) Discuss the type and stability of the critical point. 5. Consider the following linear systems and their general solutions: dx dy (a) = 6x-2y, = 4x+2y dt dt with general solution x(t) 34 (1) ) = ar ( y(t) cos(2t) + sin(2t) dx dy (b) 5x+2y, = 4x+3y dt dt 4t et +2 with general solution x(t) (360) = 0) (1 ) " + (-2) 1 dx dy (c) = 3x - Y, = 6x - 4y dt dt et x(t) y(t) with general solution 1 = 1 e- = ((1)) - 9 (6) + (1) e2t dx dy (d) = -2x - Y, = x dt dt (s sin(2t) - 2 sin(2t) - cos(2t) e4t with general solution x(t) ( ) ) = ( 11 ) y(t) For each of the linear systems e-t + 2 ( -+ + 1 ) et (i) Sketch the phase portrait near the critical point at the origin. (ii) Discuss the type and stability of the critical point.