Lipschitz condition

Nov 2009
Hi Guys,

I'm trying to get my head around this whole Lipschitz condition business, and while I understand the principle, I'm having trouble understanding it mathematically. I have the example IVP:

y'(t) = t*sqrt(|y|)

Which with y(0) = 0, fails on IVP solvers. Apparently this can be explained in terms of a Lipschitz condition, but I haven't got a clue where to start proving it.

Any help would be greatly appreciated.



MHF Hall of Honor
Mar 2009
near Piacenza (Italy)
If we write the DE as...

\(\displaystyle y^{'} = f(t,y)\) (1)

... we observe that for \(\displaystyle y=0\) the partial derivative \(\displaystyle f_{y}\) has a singularity and the the Lipschitz aren't satisfied. As consequence of that the IVP...

\(\displaystyle y^{'} = t\cdot \sqrt{|y|}\) , \(\displaystyle y(0)=0\) (2)

... has more that one solution, among them \(\displaystyle y= \frac{t^{4}}{16}\) and \(\displaystyle y = 0\)...

Kind regards

\(\displaystyle \chi\) \(\displaystyle \sigma\)
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