# Lipschitz condition

#### CorruptioN

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
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

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

• CorruptioN