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Differential Equation - Change of Variable

  1. May 27, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the general solution of

    y' = [tex] - \frac{x}{y} - \sqrt{(\frac{x}{y})^2 + 1}[/tex]

    2. Relevant equations

    3. The attempt at a solution

    I let y = ux -> y' + xu' + u

    xu' + u = - u - [tex]\sqrt{u^2+1}[/tex]

    u' = [tex]\frac{-2u - \sqrt{u^2 + 1}}{x}[/tex]

    [tex]\frac{du}{-2u - \sqrt{u^2 + 1}} = \frac{dx}{x}[/tex]

    now I'm supposed to integrate both sides, just not sure how to find the integral of [tex]\frac{du}{-2u - \sqrt{u^2 + 1}}[/tex]
  2. jcsd
  3. May 28, 2009 #2


    Staff: Mentor

    From your substitution, y = ux, it follows that u = y/x. You replaced x/y by u, instead of 1/u.

    Using the same substitution, I get
    [tex]\frac{-u du}{1 + u^2 + \sqrt{1 + u^2}} = \frac{dx}{x}[/tex]

    That's still pretty ugly on the left side, but it might be amenable to completing the square in the denominator.
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