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(a) D(D-4)y=e^(2x)f(D)y=e^(2x)The Auscillary equation is f(m)=0{:[m(m-4)=0],[m=0","m-4=0],[m=0","4],[y_(c)+c_(1)e^(0x)+c_(2)e^(4x)],[(1)/(f(D))e^(ax)=(1)/(f(a))e^(ax)],[y_(p)=p*I=(1)/(D(D-4))e^(2x)=(1)/(2(2-4))e^(2x)=(1)/(2(-2))e^(2x)=(-e^(2x))/(4)],[y_(p)=-(e^(2x))/(4)],[:.y=y_(ct)y_(p)],[y(x)=c_(1)e^(0x)+c_(2)e^(4x)-(e^(2x))/(4)" is the general solution. "],[D^(2)y+16 y=48 sin 4x],[(D^(2)+16 ... See the full answer