In an introductory course on linear differential equations I was introduced to an operator $D$ of order $n, \frac{d^n}{dx^n}$ and subsequently, all problems thereafter were solved by treating it as any other real number. For instance to solve a linear O.D.E,
$a_0(x)\frac{d^2y}{dx^2}+a_1(x)\frac{dy}{dx}+a_2(x)y=0$
$a_0(x)D^2y+a_1(x)Dy+a_2(x)y=0$
$(a_0(x)D^2+a_1(x)D+a_2(x))y=0$
$(D-m_1)(D-m_2)y=0$
The last step is not intuitive at all. How can we justify such a treatment of $D$? Can anyone explain/point me to a reference?