  The aim of this paper is twofold. First, we obtain the explicit exact formal
solutions of differential equations of different types in the form with Dyson
chronological operator exponents. This allows us to deal directly with the
solutions to the equations rather than the equations themselves. Second, we
consider in detail the algebraic properties of chronological operators,
yielding an extensive family of operator identities. The main advantage of the
approach is to handle the formal solutions at least as well as ordinary
functions. We examine from a general standpoint linear and non-linear ODEs of
any order, systems of ODEs, linear operator ODEs, linear PDEs and systems of
linear PDEs for one unknown function. The methods and techniques involved are
demonstrated on examples from important differential equations of mathematical
physics.
