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Bernoulli Differential Equation

  1. Feb 5, 2012 #1
    1. The problem statement, all variables and given/known data

    solve the given differential equation: xdy/dx + y = y^-2

    2. Relevant equations



    3. The attempt at a solution

    I don't understand how to do these substitutions...i got n=-2 then u=y^3, du/dx=3y^2

    from there i don't know where to place them
     
  2. jcsd
  3. Feb 6, 2012 #2

    vela

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    It probably doesn't make sense partially because you didn't calculate du/dx correctly. You're differentiating with respect to x, not y, so you need to use the chain rule:
    $$\frac{du}{dx} = 3y^2\frac{dy}{dx}$$

    Now, first, divide the differential equation by x so that the coefficient of y' is 1. Then get rid of the y's from the righthand side by multiplying by y2. You end up with
    $$y^2\frac{dy}{dx} + \frac{1}{x} y^3 = \frac{1}{x} $$ Now write that in terms of u=y3 and u'.
     
  4. Feb 6, 2012 #3

    HallsofIvy

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    You can also note that this is a separable equation:
    [tex]x\frac{dy}{dx}+ y= y^{-2}[/tex]
    [tex]x\frac{dy}{dx}= -y+ y^{-2}[/tex]
    [tex]\frac{dy}{-y+ y^{-2}}= \frac{dx}{x}[/tex]
    [tex]\frac{y^2 dy}{1- y^3}= \frac{dx}{x}[/tex]

    To integrate on the left, let [itex]u= 1- y^3[/itex].
     
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