  We present an approach to the canonical quantization of systems with
equations of motion that are historically called non-Lagrangian equations. Our
viewpoint of this problem is the following: despite the fact that a set of
differential equations cannot be directly identified with a set of
Euler-Lagrange equations, one can reformulate such a set in an equivalent
first-order form which can always be treated as the Euler-Lagrange equations of
a certain action. We construct such an action explicitly. It turns out that in
the general case the hamiltonization and canonical quantization of such an
action are non-trivial problems, since the theory involves time-dependent
constraints. We adopt the general approach of hamiltonization and canonical
quantization for such theories (Gitman, Tyutin, 1990) to the case under
consideration. There exists an ambiguity (not reduced to a total time
derivative) in associating a Lagrange function with a given set of equations.
We present a complete description of this ambiguity. The proposed scheme is
applied to the quantization of a general quadratic theory. In addition, we
consider the quantization of a damped oscillator and of a radiating point-like
charge.
