# Existence

• Nov 7th 2009, 01:07 PM
Nusc
Existence
u'=u^2

u(0)=1

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

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

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

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

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

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

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

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

Kind regards

$\chi$ $\sigma$
• Nov 7th 2009, 07:31 PM
Nusc
Quote:

Originally Posted by chisigma
The DE is of the type 'with separable variables' , so that its solution can be found in 'standard way' ...

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

$\chi$ $\sigma$

There shouldn't be a 2