We've been acting as though just by specifying an initial condition, there must be a solution, and it must be unique (that is, the only one corresponding to that initial condition). And, in fact, this is typically true for any ``nice'' differential equation. But which differential equations are ``nice'' enough?
We won't prove the theorem here, but we will state it. The reader is encouraged to look up the proof in any differential equations text, such as [BDH] or [HW].
In fact, the hypothesis can be weakened a little bit and still preserve the uniqueness. As long as is at least Lipshitz in , the solution will be unique. A function if called Lipschitz if there is some K so that
This means that for us, as long as we stay away from the place where our differential equation isn't defined (i.e. ensure that v > 0), we can be assured that there is a solution through every point, and that solution is unique. Notice that since our equations are autonomous (the right-hand side has no explict dependence on t), uniqueness of solutions means that solutions cannot cross in the (, v)-plane.