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Non-linear 2nd ODE involving squares of derivatives

  1. Aug 17, 2011 #1
    1. The problem statement, all variables and given/known data

    y''+(1/y)*(y')2=0

    2. Relevant equations



    3. The attempt at a solution

    This is another problem I am having trouble with. I have done searches around the internet, but seen that all "non linear" ODE of second order involves a non linear form in a non differential term (like y''+xy^2=0, or something like that), instead of the DE term.

    Punching through wolfram alpha gave a really simple straightforward answer, so I believe it shouldn't be too hard, as long as I get the general method to solve it.

    thanks in advance.
     
  2. jcsd
  3. Aug 17, 2011 #2

    dynamicsolo

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    Homework Helper

    The question to ask might be, "Where have we seen something like this:

    y'' + (1/y)*(y')2 = 0 ?"

    We can see that y can't be zero, so let's multiply through by it to get

    y y'' + (y')2 = 0 .

    Now think about the Product Rule -- what is this the derivative of? You will then be led to a separable differential equation. (And I feel like I'm talking like a fortune cookie...)
     
  4. Aug 17, 2011 #3

    ehild

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    Gold Member

    A different hint, which can be applied to all second-order equations which do not contain the independent variable x explicitly: let be the independent variable y, and denote y'=u. Then

    y''=u'=(du/dy)(dy/dx)=u (du/dy),

    and you get a first-order equation for u.

    ehild
     
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