Using the $D-$operator method and the exponential$-$shift rule $e^{ax}(D+a)f=D(e^{ax}f),$ solve the following.

$(a)\frac{d^{2}y}{d{x}^2}+5\frac{dy}{dx}-6y=3e^{-x}+x,y(0)=1,y^{\text{'}}(0)=0$

$(b)\frac{d^{2}x}{d{t}^2}-6\frac{dx}{dt}+4x=3t+2$

$(c)y^{\text{'}\text{'}}-2y^{\text{'}}-8=sin2x$

$(d)u^{\text{'}\text{'}}(t)+6u(t)=0$

$(e)\frac{d^{4}x}{d{t}^4}-81x=0$

$(f)x^{\text{'}\text{'}\text{'}}(t)+8x(t)=0$