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Separation of variables / integrating factor problem

  1. Feb 3, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the general solution for this differential equation.

    dy −2x^2 + y^2 + x
    dx = x y


    2. Relevant equations



    3. The attempt at a solution
    dy/dx = (−2x^2 + y^2 + x) / (x y)

    let y^2 = v

    dy/dx = v + x dv/dx

    v + x (dv/dx) = (-2x^2 + v^2 x^2 + x ) / v x^2

    => x (dv/dx) = (-2x^2 + v^2 x^2 + x ) / (v x^2) - (v)

    => x (dv/dx) = [ (-2x^2 + v^2 x^2 + x ) - v^2 x^2 ] / vx^2

    => x (dv/dx) = [ -2x^2 + x ] / vx^2

    => dv/dx = (-2x + 1) / vx^2

    separating variables

    v dv = [ (-2x + 1) / x^2 ] dx

    v dv = - 2(1/x) dx + (1/x^2) dx

    integrating

    (1/2)v^2 = - 2 ln x - (1/x) + c

    v^2 = -4 ln x - (2/x) + C

    substitute v = y/x

    y^2 / x^2 = -4ln x - (2/x) + C

    y^2 = -4x^2 ln(x) - 2x + Cx^2
     
    Last edited: Feb 3, 2010
  2. jcsd
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