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Math Help - Simple question in ODE

  1. #1
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    Simple question in ODE

    Given this ODE :
    y'=2*sqrt(|y|) , y(0)=0 ...
    Can we find two different soloutions around (0,0) ? If there are, find them... If there are no two different soloutions around (0,0) - explain why...

    Help is needed! TNX
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  2. #2
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    Dear WannaBe,

    I need some further details to try this problem. What do you mean by,
    y(0)=0 ?

    And what is "around (0,0)"?
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  3. #3
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    Hey there Sudharaka,
    The problem is an initial value ODE problem. i.e an equation with an initial value condition... In this one, the equation is: y'=2*sqrt(|y|) and the soloution must be a function y(x) that for x=0: y(x)=0 -> y(0)=0...It's just an initial value condition for the equation...

    When I say Around (0,0) I mean that:
    "Can we find two different soloutions in some neighborhood around (0,0) ?"

    Hope you'll be able to help me now... btw-I'll be delighted if you'll be able to help me in the other question I've posted about ODE's...

    TNX in advance...
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  4. #4
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    Dear WannaBe,
    Please see the attachment below. From that it is evident that this equation have two solutions for all x values. If you have any problems or if you can't read the attachment please feel free to contact me.

    Hope this helps.
    Attached Thumbnails Attached Thumbnails Simple question in ODE-dsc02467.jpg  
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  5. #5
    MHF Contributor chisigma's Avatar
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    An initial value problem like this ...

    y^{'} = 2\cdot \sqrt {|y|}, y(0)=0 (1)

    ... in general admits a single solution if the so called 'Lipschitz conditions' are satisfied...

    Initial value problem - Wikipedia, the free encyclopedia

    ... what is not the case of (1). If we observe the (1) we note that y^{'} is a function of the y only and that means that if \varphi(x) is solution of (1), then \varphi(x+\alpha) being \alpha an arbitrary constant is also solution of (1). In particular we can observe that solution of (1) is...

     \varphi(x) = x^{2}, x>0

    \varphi(x)=0, x\le 0 (2)

    ... so that any \varphi (x + \alpha), \alpha<0 is also solution of (1)...

    Kind regards

    \chi \sigma
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  6. #6
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    TNX a lot!
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