# Lipschitz condition

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• May 17th 2010, 04:00 PM
CorruptioN
Lipschitz condition
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.

Thanks!
• May 17th 2010, 11:41 PM
chisigma
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

$\displaystyle \chi$ $\displaystyle \sigma$