The differential equation, when used as a physical model, depend on a certain set of assumptions. For example, when we use the differential equation
$$h''(t) = -g$$
to describe the height of a thrown ball, we may set an initial height $h_0$, but the projection into the past would have the ball as coming from infinitely far below ($h = -\infty$ at $t = -\infty$), which makes no sense in real life. Clearly the ball did not surge up from below the ground, phasing through the matter, but rather was projected from your hand! The problem is that the equation has an assumption in it: namely that the only force acting is gravity, and when you project backwards you are projecting this assumption backwards. But in a likely reality, in the past, when you were getting ready to throw that ball there were other forces - namely from your hands, pockets, etc. - acting upon it, and they are not included in the equation.
The differential equation gives both a unique future and unique past history but that history, in both directions, only corresponds to reality so long as the equation's assumptions hold true, and the same also holds for the future as well, e.g. if a bird comes and knocks it while in flight then this equation won't hold either.
In your case, you cannot write down a differential equation that would be valid at all past times without knowing what assumptions are needed to correctly model the past behavior, that is, without some idea of what the past influences were. If the conditions have always been nothing has interacted with the jar then the differential equation will give that result. If that assumption is wrong, then of course, it won't work. The non-uniqueness comes out of effectively variation in the equation itself, rather than in any individual equation failing to prescribe a unique past history. You could say that in the past, the differential equation was different. If you're allowed to do that then of course multiple past histories can end up with the same starting point - the theorem they don't is dependent on using the same equation to go backward!