Find general solution of the system x Ax where A and sketc

Find general solution of the system x\' = Ax, where A = and sketch the phase portrait. (Please try to be more precise.)

Solution

x\'= -x-2y ==> 2y= -x- x\' ==> y= ( -x\'-x)/2 ==> y\'=( -x\'\'-x\')/2 y\'= 4x+3y ( -x\'\'-x\')/2 = 4x + 3 ( -x\'-x)/2 -x\'\'-x\' = 8x -3x\' -3x x\'\' - 2x\' +5x=0 m^2 -2m +5=0 m= 1-2i m=1+2i x(t) = e^(t) ( A cos(2x) + B sin(2x) ) y= ( -x\'-x)/2 ==> y(t) = ...