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Calculate differential equation

  1. Feb 14, 2014 #1
    1. The problem statement, all variables and given/known data

    [tex]y=xy^\prime-\left(y^\prime\right)^2[/tex]

    2. Relevant equations



    3. The attempt at a solution

    Unfortunately, I do not have any good idea. I tried [tex]y=xt(x)[/tex], but the equation only became worse.
     
  2. jcsd
  3. Feb 14, 2014 #2

    pasmith

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    Homework Helper

    Differentiate both sides with respect to [itex]x[/itex]. You do have to check whether all the solutions of the resulting 2nd order ODE are solutions of this ODE.
     
  4. Feb 14, 2014 #3
    Very clever :)

    [tex]y^\prime=y^\prime+xy^{\prime\prime}-2y^\prime y^{\prime\prime}[/tex]

    [tex]xy^{\prime\prime}-2y^\prime y^{\prime\prime}=0[/tex]

    [tex]y^{\prime\prime}\left(x-2y^\prime\right)=0[/tex]

    Now we could expect that [tex]y=ax+b[/tex], but:

    [tex]y^\prime=a[/tex]

    [tex]ax+b=ax-a^2[/tex]

    [tex]b+a^2=0[/tex]

    The last equation must be satisfied in order to be a solution. And of course second solution: [tex]y=\frac{x^2}{4}+C[/tex]

    Is it good answer?
     
  5. Feb 14, 2014 #4

    pasmith

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    This is correct.

    Is there any restriction on the value of [itex]C[/itex]?
     
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