
IVP
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:
$\displaystyle x'(t)=\frac{t}{1x}, x(0)=2$
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:
$\displaystyle x(t)=1+\sqrt{1t^2}$
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!
marine