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Math Help - Existence

  1. #1
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    Question Existence

    u'=u^2

    u(0)=1

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

    hint?
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  2. #2
    MHF Contributor chisigma's Avatar
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    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
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  3. #3
    Junior Member
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    Quote Originally Posted by chisigma View Post
    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
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