Lipschitz condition

Nov 2009
10
0
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!
 

chisigma

MHF Hall of Honor
Mar 2009
2,162
994
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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