A mass of 1 slug when attached to a spring whose constant is

A mass of 1 slug, when attached to a spring whose constant is 5 lb/ft. Initially the mass is released 1 foot below the equilibrium position with a downward velocity of 5 ft/s and the subsequent motion takes place in a medium that offers a damping force that is numerically equal to 2 times the instantaneous velocity.

Find the equation of motion if the mass is driven by an external force equal to f(t) = 3sin2t.

Solution

mx\'\'=-kx-bx\'+f(t)
m=1
k=2
b=5
x\'\'+2x\'+5x=f(t)
x\'\'+2x\'+5x=3sin2t

X(t)=Xc(t) + Xp(t)
= e^-t(c1cos2t+c2sin2t)+(Asin2t+Bsin2t
Xp(t)= Acos2t+Bsin2t
Xp\'(t)=-2Asin2t+2Bcos2t
Xp\'\'(t)=-4cos2t-4Bsin2t

putting these value we get
(-4A+4B+5A)cos2t + (-4B-4A+5B)sin2t=3sin2t

A+4B=0
-4A+B=3

solving we get A=-12/17 B=3/17