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

|(, *t*) - (, *t*)| < *K*| - |

for all

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.

2002-08-29