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Another ODE, can't separate variables

  1. Jun 14, 2008 #1
    This is in a problem set for variables separate but I can't seem to separate them, and I do not know how to proceed.

    (x^2)dy + 2xy dx = (x^2) dx

    The solution given is: (3x^2)y = x^3 + c
     
  2. jcsd
  3. Jun 14, 2008 #2

    rock.freak667

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    [tex]x^2dy + 2xy dx = x^2 dx[/tex]

    [tex]\equiv x^2 \frac{dy}{dx}+2xy=x^2[/tex]

    [tex]\equiv \frac{dy}{dx}+\frac{2}{x}y=1[/tex]


    this is in a form that you should know how to solve and it isn't a separation of variables type.

    If you don't know how to solve ODE's in this form check this link Integrating Factor.
     
  4. Jun 15, 2008 #3
    Thank you, yes I can solve from here. But I still don't understand why this was in the separation of variables chapter's problem set.
     
  5. Jun 15, 2008 #4

    arildno

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    Well, introduce the new variable:
    [tex]u=\frac{y}{x}[/tex]
    Then, we have:
    [tex]\frac{du}{dx}=\frac{1}{x}\frac{dy}{dx}-\frac{u}{x}\to\frac{dy}{dx}=x\frac{du}dx}+u[/tex]

    We therefore get the diff.eq:
    [tex]x\frac{du}{dx}+u+2u=1\to\frac{1}{1-3u}\frac{du}{dx}=\frac{1}{x}[/tex], which is separable.
     
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