  Three approaches are considered to solve the equation which describes the
time-dependent diffusive shock acceleration of test particles at the
non-relativistic shocks. At first, the solution of Drury (1983) for the
particle distribution function at the shock is generalized to any relation
between the acceleration time-scales upstream and downstream and for the
time-dependent injection efficiency. Three alternative solutions for the
spatial dependence of the distribution function are derived. Then, the two
other approaches to solve the time-dependent equation are presented, one of
which does not require the Laplace transform. At the end, our more general
solution is discussed, with a particular attention to the time-dependent
injection in supernova remnants. It is shown that, comparing to the case with
the dominant upstream acceleration time-scale, the maximum momentum of
accelerated particles shifts toward the smaller momenta with increase of the
downstream acceleration time-scale. The time-dependent injection affects the
shape of the particle spectrum. In particular, i) the power-law index is not
solely determined by the shock compression, in contrast to the stationary
solution; ii) the larger the injection efficiency during the first decades
after the supernova explosion, the harder the particle spectrum around the
high-energy cutoff at the later times. This is important, in particular, for
interpretation of the radio and gamma-ray observations of supernova remnants,
as demonstrated on a number of examples.
