We cannot say that. My point is that the existence/uniqueness theorem was only proven for linear differencial equations. When we have non-linear differencial equations we have no idea whether it works or not without solving it.

The theorem says that given,

With

.

Where

are continous functions on some open interval

and where

then there exists exactly one solution to that differencial equation.

So given (after division),

.

We must work on the interval containing

because if you look at the theorem the initial point for the equation is contained in the interval. But that is not it. We also require that

and

be continous. Now,

is not continous at certain points (graph it). But if it needs to contain zero so the only open interval (largest interval) where

is continous is on

. But we also require that

be continous, so we need to exclude the point

. This point

is contained in

we we need to omit it. This gives us that on

these coefficient functions are continous. So there is a unique solution.