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Homework Help: Find all solutions?

  1. Oct 8, 2014 #1
    1. The problem statement, all variables and given/known data
    Find all solutions of xy'=x^3+(1-2x^2)y+xy^2.

    2. Relevant equations

    3. The attempt at a solution
    Here's my work:

    Now I'm stucked. Please help me.
  2. jcsd
  3. Oct 8, 2014 #2


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    Homework Helper

    You can see from the last line that [itex]y(x) = x[/itex] is one solution, although there may be others. But your rearrangement is not separable, so you are unlikely to make further progress.

    The left hand side of the original is [itex]xy'[/itex]. There's a [itex]y[/itex] on the right, so bringing that across makes the LHS [itex]xy' - y = x^2(y/x)'[/itex], so the substitution [itex]v = y/x[/itex] is worth considering.
  4. Oct 9, 2014 #3


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    Science Advisor

    Good idea!

    If v= y/x, then y= xv so that y'= xv'+ v. xy'=x^3+y-2x^2*y+xy^2 becomes x^2v'+ xv= x^3+ xv- 2x^3v+ x^3v^2.
  5. Oct 9, 2014 #4
    Thank you so much for the help, Hallsoflvy.
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