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Help with solving this differential equation?

  1. Sep 18, 2012 #1
    1. The problem statement, all variables and given/known data

    (3y4-11x2y2-28x4)dx-(4xy3)dy=0

    2. Relevant equations

    My=Nx if equation is exact, if not, I can make it exact (I hope) by finding an integrating factor. The problem is I can't get the integrating factor to be a function of x or y only.

    3. The attempt at a solution

    Let stuff in front of dx=M
    Let stuff in front of dy=N

    My=12y3-22x2y
    Nx=-4y3
    My-Nx=16y3-22x2y

    First try (My-Nx)/N=(16y3-22x2y)/4xy3

    Not a function of x only.

    Then (My-Nx)/-M=16y3-22x2y/-(3y4-11x2y2)

    Not a function of y only.

    Am I using the wrong method to solve this question? I don't see any other way to go about it rather than turning this into an exact equation. Why can't I find an integrating factor that has a pure x or y expression?

    Thanks.
     
  2. jcsd
  3. Sep 18, 2012 #2

    LCKurtz

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    M(x,y) and N(x,y) are both homogeneous of degree 4. So y = ux will make a separable equation of it.
     
  4. Sep 18, 2012 #3
    Sorry, I haven't quite learned that yet. Do you mind elaborating a little? Is that another technique for solving differential equations? Something called substitution, maybe?

    Thank you
     
  5. Sep 18, 2012 #4

    LCKurtz

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    A function ##f(x,y)## is homogeneous of degree ##n## if ##f(\lambda x,\lambda y)
    =\lambda^nf(x,y)##. What ##n## works for your ##M## and ##N##?

    Try the substitution ##y = ux,\quad dy =xdu + udx##.

    [Edit, added later]: I see I forgot to tell what this type of DE is called. If is called a first order homogeneous equation. Unfortunately, this is not the same meaning of the term as when applied to a linear DE with 0 on the right side, where we refer to homogeneous vs. non-homogeneous equations. In the setting of your problem, it refers to the ##M## and ##N## being homogeneous in the above sense.
     
    Last edited: Sep 18, 2012
  6. Sep 18, 2012 #5
    [solved]
     
    Last edited: Sep 18, 2012
  7. Sep 18, 2012 #6
    Nevermind I made a mistake, that's why I thought I couldn't turn the last one into an exact one.

    Thank you once again! :)
     
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