Homework SourseFind general solution of the system x Ax where A 1 2 1 4 a
Find general solution of the system x\' = Ax,
where A = 1 2
-1 4
and sketch the phase portrait. (Please try to be more pre-
cise.)
where A = 1 2
-1 4
and sketch the phase portrait. (Please try to be more pre-
cise.)
Solution
x\'= x+2y ==> 2y= x\'-x ==> y=(x\'-x)/2 ==> y\'=(x\'\'-x\')/2 y\'= -x+ 4y (x\'\'-x\')/2 = -x + 4 (x\'-x)/2 x\'\' -x\' = -2x +4x\' -4x x\'\' -5x\' +6x=0 m^2-5m +6=0 (m-2)(m-3)=0 m=2 m=3 x(t) = A e^(2t) + Be^(3t) AND x\' = 2A e^(2t) + 3Be^(3t) y=(x\'-x)/2 ==> y(t) = (2A e^(2t) + 3Be^(3t) - (A e^(2t) + Be^(3t)) )/2 y(t) = ( - A e^(2t) + 2 Be^(3t))/2 THUS x(t) = A e^(2t) + Be^(3t) y(t) = ( - A e^(2t) + 2 Be^(3t))/2
