Solution to Linear Nonhomogeneous Differential Equation

Solution to Linear Nonhomogeneous Differential Equation

Consider system:
$x'(t) = 3x(t) +e^(3t)$
$y'(t) = 2x(t) -y(t) -2z(t)$
$z'(t) = 3x(t) +6y(t) +6z(t)$
by first finding 3 lin. indep. solutions to the homogeneous Vector ODE:
$x'(t) = C x(t)$
then constructing the fundamental matrix $M(t|0)$
and hence finding the solution for $x(t)$ , $y(t)$ , $z(t)$ given ,
$x(0)=1$, $y(0)=0$, $z(0)=0$