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How do i solve this ODE?

  1. Sep 17, 2014 #1
    1. The problem statement, all variables and given/known data
    the first one
    y'=[itex]\frac{y^{2}+xy^{2}}{x^{2}y-x^{2}}[/itex]

    the second one
    xyy'=[itex]\frac{x^{2}+1}{y+1}[/itex]
    2. Relevant equations



    3. The attempt at a solution
    i separated x and y variable then integrate both of them

    in the first one
    ∫[itex]\frac{y-1}{y^{2}}[/itex]dy=∫[itex]\frac{1+x}{x^{2}}[/itex]dx

    ln|y|+[itex]\frac{1}{y}[/itex]+C=- [itex]\frac{1}{x}[/itex]+ln|x|+C

    and the second one
    ∫y(y+1)dy = ∫[itex]\frac{x^{2}+1}{x}[/itex]dx

    [itex]\frac{y^{3}}{3}[/itex]+[itex]\frac{y^{2}}{2}[/itex]+C=[itex]\frac{x^{2}}{2}[/itex]+ln|x|+C

    but i can't change both of them into f(x) form or any simpler form
     
  2. jcsd
  3. Sep 17, 2014 #2
    It is rare that you will find a differential equation with a solution that can be written as an explicit function. Implicit solutions, the equations relating x and y that you found, are usually accepted as finding a solution to a differential equation as well. As long as there are no derivatives in your final equation, and you specify the domain of the implicit function y that is defined by your equation, where it satisfies the original differential equation, you have found a solution.
    Note, however, that you do not need two constants of integration: you may condense them into a single constant: C1 - C2 = C.
     
  4. Sep 18, 2014 #3
    i see, i just don't really understand the difference between implicit and explicit form, so the thing i just solve is the implicit form.. thanks for answering
     
  5. Sep 19, 2014 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    The only thing to "understand" about "implicit" and "explicit" form is that the explicit form is always "y= some expression in x only" and the implicit form isn't!
     
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