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How to solve this ODE?

  1. Oct 22, 2011 #1
    [itex][/itex]1. The problem statement, all variables and given/known data
    [tex]2y(1+x^2\sqrt{y})dx + x(2+x^2\sqrt{y})dy = 0[/tex]

    3. The attempt at a solution
    well, I substituted x^2√y=u but then when I tried to differentiate it I understood it would be so hard. Please check and see if I've differentiated it correctly:

    √y = u/x^2 -> y = u^2.x^-4 -> dy/dx = 2u.u'x^(-4) - 4x^(-5).u^2
    Is that correct? if yes, then I think I've just made the problem harder. how can I solve that ODE?
     
  2. jcsd
  3. Oct 22, 2011 #2

    ehild

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    Homework Helper
    Gold Member

    It is correct. Substitute for y and y' in the original equation, simplify and arrange: it is a separable ODE.

    ehild
     
  4. Oct 22, 2011 #3

    phyzguy

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    Science Advisor

    You could try grouping the terms into perfect differentials. For example, 2 y dx + 2 x dy = 2 d(xy). The other two terms can be written similarly as d(something) . Then write u = xy and v = something and it is a simple linear ODE.
     
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