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Diffrential equations, integration factor with two vars

  1. Mar 27, 2015 #1
    1. I need to find a condition that the equation will have a integration factor from the shape K(x*y).
    (K-integration factor sign)


    2.the eq from the shape M(x,y)dx+N(x,y)dy=0 ,not have to be exact!


    3. i tried to open from the basics. d(k(x*y)M(x,y))/dy=d((k(x*y)N(x,y))/dx.
    and i used the fact that d(k(x,y))/dy is x (exc. for dx)/

    im hoping i was clear enough , thx and sorry for my bad english.
     
  2. jcsd
  3. Mar 27, 2015 #2

    BiGyElLoWhAt

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    well, normally an integration factor will be of the form ##e^{\int P(x)}##
    I'm confused by your notation, it seems as though you're setting up an exact function, or maybe that's what you're saying in 2: it's not an exact function?
     
  4. Mar 27, 2015 #3

    Mark44

    Staff: Mentor

    "the equation" -- What equation?
    It's not clear to me at all. What is the equation you're trying to solve?
     
  5. Mar 27, 2015 #4

    LCKurtz

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  6. Mar 27, 2015 #5

    HallsofIvy

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    The initial equation is M(x,y)dx+ N(x,y)dy= 0 and you multiply by k(xy) where k is some function of a single variable: k(xy)M(x, y)dx+ k(xy)N(x,y)dy= 0.
    In order that this be "exact" we must have [itex](k(xy)M(x, y))_y= (k(xy)N(x, y))_x[/itex].

    xk'(xy)M(x,y)+ k(xy)M_y(x, y)= yk'(x,y)N(x,y)+ k(xy)N_x(x,y)

    (xM(x,y)- yN(x,y))k'(xy)= k(xy)(N_x(x,y)- M_y(x,y)
     
  7. Mar 29, 2015 #6
    thanks a lot guys u all helped me
     
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