u'=u^2

u(0)=1

find $\displaystyle \delta$>0 s.t. the solution exists $\displaystyle [-\delta,\delta]$

hint?

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- Nov 7th 2009, 01:07 PMNuscExistence
u'=u^2

u(0)=1

find $\displaystyle \delta$>0 s.t. the solution exists $\displaystyle [-\delta,\delta]$

hint? - Nov 7th 2009, 07:05 PMchisigma
The DE is of the type 'with separable variables' , so that its solution can be found in 'standard way' ...

$\displaystyle \frac{du}{dt} = u^{2} \rightarrow \frac{du}{u^{2}} = dt \rightarrow \frac{2}{u} = c - t$ (1)

... and taking into account the 'initial conditions' we have...

$\displaystyle u= \frac{1}{1-\frac{t}{2}}$ (2)

It is evident from (2) that u(*) has a singularity in $\displaystyle t=2$ and this fact is confirmed if we expand u(*) in series around $\displaystyle t=0$ ...

$\displaystyle u(t) = \sum_{n=0}^{\infty} (\frac{t}{2})^{n}$ (3)

... which converges for $\displaystyle |t|<2$. So the value of $\displaystyle \delta$ You are searching is $\displaystyle \delta=2$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$ - Nov 7th 2009, 07:31 PMNusc