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Homework Help: Integrating factor ode

  1. Apr 5, 2010 #1
    1. The problem statement, all variables and given/known data

    Solve:
    [tex](1-\frac{x}{y})dx + (2xy + \frac{x}{y} + \frac{x^2}{y^2})dy = 0[/tex]

    3. The attempt at a solution

    No idea what strategy to use here. Tried using an integrating factor, but no success. A lot of x/y in here makes me think I need to use a substitution, but there's also "xy" in here which doesn't help me. What should I do?
     
  2. jcsd
  3. Apr 5, 2010 #2
    Re: Ode

    How did you try using the integrating factor? Did you try multiplying by "x^m*y^n" and then solving the integers "m" and "n"?
     
  4. Apr 5, 2010 #3
    Re: Ode

    I tried using the formulas:

    [tex]
    F(x) = \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N}
    [/tex]
    And then:
    [tex]\mu(x) = e^{\int F(x) dx}[/tex]

    (there's also a similar one for y, with signs reversed and M instead of N in the denominator).

    Had trouble in the F(x) part - couldn't get a function of x only (or y, for that matter).
     
    Last edited: Apr 5, 2010
  5. Apr 5, 2010 #4
    Re: Ode

    Ok! Solved it. Missed a little thing on my side.

    Thanks anyway! :)
     
  6. Apr 5, 2010 #5

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    Re: Ode

    The substitution [itex]u(y)=\frac{x}{y}[/itex] works out nicely.
     
  7. Apr 5, 2010 #6
    Re: Ode

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

    That's the solution, in 2 different strategies.
    One using the integrating factor, second using your substitution. Thanks for pointing it out. :)
     
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