By the method of undetermined coefficients trial and error m

By the method of undetermined coefficients (trial and error method), find the steady state solution (alter transients have died away) to

Solution

put x1 = A1cos3t + B1sin3t and x2 = A2cos3t + B2sin3t

equations become,
-3A1sin3t + 3B1cos3t = -(A1cos3t + B1sin3t) + 2(A2cos3t + B2sin3t)
and
-3A2sin3t + 3B2cos3t = -3(A1cos3t + B1sin3t) - 2(A2cos3t + B2sin3t) + cos3t

=>
-3A1sin3t + 3B1cos3t = (2A2-A1)cos3t + (2B2 - B1)sin3t
and
-3A2sin3t + 3B2cos3t = (1 -2A2 -3A1)cos3t + (-2B2 -3B1)sin3t

=>
-3A1 = 2A2 - A1 and 3B1 = 2B2 - B1
and
-3A2 = 1 - 2A2 -3A1 and 3B2 = -2B2 - 3B1

=>A1 = 1/4 ,B1 = 0
and A2 = -1/4, B2 = 0

so,

x1 = 0.75cos3t
x2 = -0.75cos3t