Hello there!

I'm having some trouble with the existence and uniqueness (E&U) thms for ODEs:

Consider the IVP:

$\displaystyle x'(t)=f(t,x(t)),f(\xi)=\eta$

In order to use some Existence and Uniqueness(E&U) thms we need f to be Lipschitz or at least locally Lipschitz in x.

The problem is now how to prove this in special cases.

Consider the special case:

$\displaystyle x'(t)=\frac{t}{1-x}, x(0)=2$

Now, assuming we want to prove existence and uniqueness we must first find the domain of f, i.e.

$\displaystyle f: R\times R-\{1\} \longrightarrow R$
$\displaystyle (t,x)\longrightarrow f(t,x)$

but I couldn't estimate f to have any Lipschitz continuity because of the singularity at 1

If we solve the equation - we find the solution to be:

$\displaystyle x(t)=1+\sqrt{1-t^2}$

which sets another domain for f: (-1,1)x(1,2]->R, apparently the more precise one.

How should I proceed in such cases?
Also, consider cases where you have to prove E&U but cannot solve the equation explicitly. Then what?!

thanks in advance!