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Integrating factors: How to turn 3xy^3 + (1+3x^2y^2)dy/dx=0 to dy/dx+P(x)=Q(x) form

  1. Dec 10, 2009 #1
    1. The problem statement, all variables and given/known data
    Solve 3xy^3 + (1+3x^2y^2)dy/dx=0 using integrating factors


    2. Relevant equations
    y' + p(x) = q(x)


    3. The attempt at a solution
    I'm having trouble putting the equation to y' + p(x) = q(x)
    I distributed dy/dx so it becomes 3xy^3dy/dx + 1dy/dx+3x^2y^2dy/dx=0
    But I didn't know where to go from there.
    So I multiplied both sides by dx and 3xy^3dx + (1+3x^2y^2)dy=0
    I don't know how to start this, please help!
     
  2. jcsd
  3. Dec 10, 2009 #2
    Re: Integrating factors: How to turn 3xy^3 + (1+3x^2y^2)dy/dx=0 to dy/dx+P(x)=Q(x) fo

    For starters, it's y' + P(x)*y = q(x)

    For this to be true, the DE has to be linear.

    Do you think it is linear, separable or neither?
     
  4. Dec 10, 2009 #3
    Re: Integrating factors: How to turn 3xy^3 + (1+3x^2y^2)dy/dx=0 to dy/dx+P(x)=Q(x) fo

    the definition i have for a linear DE is that it is a DE that can be written in the form y' + P(x)*y = q(x). I am trying to rewrite the DE in that form, but it looks like I can't. If I can't, then according to the definition I have, the equation is not linear, and therefore not separable. But there is an answer from the book's answer set, so it looks like it should be linear...
     
  5. Dec 10, 2009 #4
    Re: Integrating factors: How to turn 3xy^3 + (1+3x^2y^2)dy/dx=0 to dy/dx+P(x)=Q(x) fo

    You won't need to rely upon integrating factors in this case.

    we know dy/dx = -3xy^3/(1 + 3x^2y^2)

    Thus: dx/dy = -1/3xy^3 - x/y

    Making a simple substitution of u = xy

    dx/dy = (y*du/dy - u)/y^2 when the substitution is made

    The equation should become separable.
     
  6. Dec 11, 2009 #5
    Re: Integrating factors: How to turn 3xy^3 + (1+3x^2y^2)dy/dx=0 to dy/dx+P(x)=Q(x) fo

    ohhh, i see it now. thank you!
     
  7. Dec 11, 2009 #6

    HallsofIvy

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    Re: Integrating factors: How to turn 3xy^3 + (1+3x^2y^2)dy/dx=0 to dy/dx+P(x)=Q(x) fo

    Why did you say "the equation is not linear, and therefore not separable"? Most separable equations are not linear. An easy example is dy/dx= x/y.
     
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