I think we can assume (via the Picard existence theorem, like you mention) that given an (ordinary) differential equation:
Where
is analytic (actually this is a stronger condition than required, but it's all we need here), then specifying a set of initial conditions:
Singles out a
unique solution to the equation (at least in some neighborhood of
). (Where
,... etc. are constants).
Given that, it's a simple matter to show that for the general solution
we can, by adjusting A and B, set
and
to any values we choose, for any given
. Thus, the above general solution includes
every possible solution, and so is the most general possible.
