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Answer 1 · 2017-12-24T11:01:14

See below.

Assuming null initial conditions and using the Laplace transformation

#(s^2-4s+1)Y(s) = 2/((s-2)^2+2^2)# or

#Y(s) = 2/((s^2-4s+1)((s-2)^2+2^2)) = A/(s-2+sqrt3)+B/(s-2-sqrt3)+(sC+D)/((s-2)^2+2^2)#

then

#y(x) = (A e^((2+sqrt3)x)+B e^((2-sqrt3)x)+C(e^(2x)(cos(2x)+sin(2x))+D e^(2x)cosx sinx)h(x)#

#h(x)# is the unit step function.

The constants #A,B,C,D# are obtained by residue calculation, giving

#{(A=97/(7 sqrt3)-8),(B=-97/(7 sqrt3)-8),(C=0),(D=-128/7):}#