[tex]\frac{dy}{dx}=f(x,y);\qquad y(a)=b[/tex]

And the extension of this to higher-ordered equations.

I'd like to understand the sufficient and necessary conditions for uniqueness.

Most proofs require that f(x,y) and its partial with respect to y be continuous. However, Ince shows that continuity of f(x,y) is not a necessary condition for uniqueness. As I read his analysis, it appears that the one necessary condition for uniqueness is that f(x,y) MUST satisfy a Lipschitz condition (or a condition of a similar nature) about the initial point [itex]x_0[/itex]:

If [itex](x,y_1)[/itex] and [itex](x,y_2) [/itex] be any points about some defined region containing [itex]x_0[/itex], then:

[tex]|f(x,y_1)-f(x,y_2)|<K|y_1-y_2|[/tex]

for some constant K.

It's still a bit unclear to me.

May I ask this: If f(x,y) does NOT satisfy a Lipschitz condition in a neighborhood of [itex]x_0[/itex], then no unique solution exists for the differential equation and if a unique solution exists, then f(x,y) must necessarilly satisfy a Lipschitz condition?