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.