  Let $\mathcal{O}\subset\mathbb{R}^d$ be a bounded domain of class $C^{1,1}$.
In $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a selfadjoint matrix second
order elliptic differential operator $B_{D,\varepsilon}$,
$0<\varepsilon\leqslant1$, with the Dirichlet boundary condition. The principal
part of the operator is given in a factorized form. The operator involves first
and zero order terms. The operator $B_{D,\varepsilon}$ is positive definite;
its coefficients are periodic and depend on $\mathbf{x}/\varepsilon$. We study
the behavior of the operator exponential $e^{-B_{D,\varepsilon}t}$, $t>0$, as
$\varepsilon\rightarrow 0$. We obtain approximations for the exponential
$e^{-B_{D,\varepsilon}t}$ in the operator norm on
$L_2(\mathcal{O};\mathbb{C}^n)$ and in the norm of operators acting from
$L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space
$H^1(\mathcal{O};\mathbb{C}^n)$. The results are applied to homogenization of
solutions of the first initial boundary-value problem for parabolic systems.
