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Supposedly simple calculus problem.

  1. Nov 2, 2011 #1
    1. The problem statement, all variables and given/known data

    Solve the differential equation.

    [itex]\frac{dy}{dx}[/itex] = [itex]\frac{1}{xy}[/itex] + [itex]\frac{y}{x}[/itex]


    3. The attempt at a solution

    [itex]\frac{dy}{dx}[/itex] = [itex]\frac{x + xy^{2}}{x^{2}y}[/itex]

    [itex]\frac{dy}{dx}[/itex] = ([itex]\frac{x}{x^{2}}[/itex])([itex]\frac{1 + y^{2}}{y}[/itex])

    ([itex]\frac{y}{1 + y^{2}}[/itex])dy = [itex]\frac{x}{x^{2}}[/itex]dx

    [itex]\oint\frac{y}{1 + y^{2}}[/itex]dy = [itex]\oint\frac{x}{x^{2}}[/itex]dx

    I can't seem to get any further than there
     
    Last edited: Nov 2, 2011
  2. jcsd
  3. Nov 2, 2011 #2

    lurflurf

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    change variables u=y/x
     
  4. Nov 2, 2011 #3

    Mark44

    Staff: Mentor

    It's simpler to just factor out 1/x, giving
    dy/dx = (1/x)(1/y + y) = (1/x)(1 + y2)/y
    The first integral can be done with an ordinary substitution. In the integral on the right, simplify x/x2.
     
  5. Nov 2, 2011 #4

    Ray Vickson

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    Try multiplying both sides of your original DE by y.

    RGV
     
  6. Nov 2, 2011 #5

    Mark44

    Staff: Mentor

    Guys, the equation is separable, and the OP is close to getting a solution. IMO, the best thing to do is help him along the path he's on (which is the simplest), rather than steering him in a different direction.
     
  7. Nov 2, 2011 #6

    Ray Vickson

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    I could not see the rest of his submission on my i-phone at the coffee shop, so that's why I replied.

    RGV
     
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